Browse by properties

Find special flows by their ambient space, topology, embeddedness, and Gaussian area or entropy. Filters are combined within the same dimensional case of a family.

Conventions and how to read the data

Examples follow the chapter order: category, lowest recorded ambient dimension, then earliest recorded reference, including first arXiv submissions. Dates concern the references recorded in this catalogue, not necessarily original discovery. Parenthetical author groups credit each cited paper in chronological order.

Topology refers to the parameter manifold for an immersion and to a regular time slice for an evolving flow. Surface genus applies only in dimension two. Ends are topological ends; several asymptotic sheets in one height direction need not be different ends. For disconnected examples the text states whether a count is per component. Ambient space means the space stated in the construction, rather than every possible Euclidean inclusion.

Embeddedness describes the stated immersion or regular time slices in the full ambient space, not a projection or a singular or infinite-time limit. “Immersed” alone does not mean “not embedded.” Multiple covers are not embeddings. “Depends on the construction” identifies families whose status depends on their factors or input immersion; these do not match either definite filter. “Not recorded” means no conclusion is recorded here. Networks are called embedded when their branches meet only at the prescribed junctions; they are not smooth curves at those junctions.

For an n-dimensional Euclidean submanifold, including immersion multiplicity,

\[F_{x_0,\tau}(M)=(4\pi\tau)^{-n/2}\int_M e^{-|X-x_0|^2/(4\tau)}\,d\mu,\qquad \lambda(M)=\sup_{x_0,\,\tau>0}F_{x_0,\tau}(M).\]

Centered Gaussian area means F₀,₁ of the stated slice. It generally differs from entropy and depends on center and scale. For the normalized complete shrinkers with polynomial area growth here, F₀,₁ equals entropy. A bound on the supremum over a flow's times is labeled separately from the entropy of a single slice. Euclidean entropy is not assigned to an intrinsic flow in a curved ambient manifold.

“Not recorded” means this catalogue has no value or theorem recorded, not that none is known. The proved-entropy-bound filter uses recorded scalar values and upper bounds; a parameter-dependent formula without a recorded uniform bound will not match. Numerical estimates are excluded. A family may appear in several groups. References and hypotheses are in the linked entries.

154 example families

Closed planar shrinking curves (Abresch–Langer, 1986; Halldórsson, 2010; Drugan–Lee–X. H. Nguyen, 2017; Colding–Minicozzi II, 2019)

Numerical ODE profile

Earliest recorded reference: 1986

Immersed circle S¹ · Euclidean plane ℝ² · Not embedded

Ambient space: Euclidean plane ℝ²
Topology: Immersed circle S¹
Embeddedness: Not embedded: The noncircular primitive Abresch–Langer curves recorded here have self-intersections. The embedded shrinking circle is listed separately.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Proved bound or relation: λ > 2 for the noncircular closed curves.
Gaussian area: Proved bound or relation: F₀,₁ = λ for the normalized shrinker. λ > 2 for the noncircular closed curves.

Spoon-shaped shrinking network (Chen–Guo, 2007; Chang, 2021)

Numerical ODE network

Earliest recorded reference: 2007

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Regular planar network · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: One bounded loop attached to a straight ray at a single triple junction
Embeddedness: Embedded: Embedded as a network with prescribed 120-degree triple junctions, not a smooth curve there.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: One topological end.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Shrinking lens network (Chen–Guo, 2007; Chang, 2021)

Numerical ODE network

Earliest recorded reference: 2007

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Regular planar network · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Two curved arcs bounding a lens and two exterior rays
Embeddedness: Embedded: Embedded as a network with prescribed 120-degree triple junctions, not a smooth curve there.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Two topological ends.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Shrinking fish network (Chen–Guo, 2007; Chang, 2021)

Source-informed network schematic

Earliest recorded reference: 2007

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Regular planar network · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: The asymmetric two-ray, one-region configuration
Embeddedness: Embedded: Embedded as a network with prescribed 120-degree triple junctions, not a smooth curve there.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Two topological ends.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Three-ray shrinking star (Chen–Guo, 2007; Chang, 2021)

Numerical ODE network

Earliest recorded reference: 2007

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Regular planar network · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: One bounded three-sided region and three exterior rays
Embeddedness: Embedded: Embedded as a network with prescribed 120-degree triple junctions, not a smooth curve there.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Three topological ends.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Four-ray shrinking star (Chen–Guo, 2007; Chang, 2021)

Numerical ODE network

Earliest recorded reference: 2007

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Regular planar network · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: One bounded four-sided region and four exterior rays
Embeddedness: Embedded: Embedded as a network with prescribed 120-degree triple junctions, not a smooth curve there.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: At least four ends, as specified in the statement.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Five-ray shrinking star (Chen–Guo, 2007; Chang, 2021)

Numerical ODE network

Earliest recorded reference: 2007

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Regular planar network · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: One bounded five-sided region and five exterior rays
Embeddedness: Embedded: Embedded as a network with prescribed 120-degree triple junctions, not a smooth curve there.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: At least four ends, as specified in the statement.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Shrinking rocket network (Chen–Guo, 2007; Chang, 2021)

Source-informed network schematic

Earliest recorded reference: 2007

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Regular planar network · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: The asymmetric three-ray, one-region configuration
Embeddedness: Embedded: Embedded as a network with prescribed 120-degree triple junctions, not a smooth curve there.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Three topological ends.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Round shrinking spheres (Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017)

Exact formula

Earliest recorded reference: 2009-08-26

Shrinking circle · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Circle S¹
Embeddedness: Embedded: The round spheres, including the planar circle, are embedded hypersurfaces.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Exact formula: λ = √(2π/e) ≈ 1.52035.
Gaussian area: Exact formula: F₀,₁ = λ for the normalized shrinker. λ = √(2π/e) ≈ 1.52035.

Round two-sphere · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Sphere S²
Embeddedness: Embedded: The round spheres, including the planar circle, are embedded hypersurfaces.
Compactness: Compact
Surface genus: 0
Topological ends: 0 (compact)
Entropy: Exact formula: λ = 4/e ≈ 1.47152.
Gaussian area: Exact formula: F₀,₁ = λ for the normalized shrinker. λ = 4/e ≈ 1.47152.

Higher-dimensional spheres · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sphere Sⁿ, n ≥ 3
Embeddedness: Embedded: The round spheres, including the planar circle, are embedded hypersurfaces.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Exact formula: λ = (4π)^(−n/2) (2n)^(n/2) e^(−n/2) |Sⁿ(1)|; depends on n.
Gaussian area: Exact formula: F₀,₁ = λ for the normalized shrinker. λ = (4π)^(−n/2) (2n)^(n/2) e^(−n/2) |Sⁿ(1)|; depends on n.

Planes through the shrinker center (Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017)

Exact formula

Earliest recorded reference: 2009-08-26

Lines in ℝᴺ, N ≥ 2 · Euclidean plane ℝ²; Euclidean space ℝ³; Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean plane ℝ²; Euclidean space ℝ³; Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Line ℝ
Embeddedness: Embedded: The linear-subspace inclusion is an embedding.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2
Entropy: Exact formula: λ = 1.
Gaussian area: Exact formula: F₀,₁ = λ for the normalized shrinker. λ = 1.

Planes in ℝᴺ, N ≥ 3 · Euclidean space ℝ³; Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ³; Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Plane ℝ²
Embeddedness: Embedded: The linear-subspace inclusion is an embedding.
Compactness: Noncompact
Surface genus: 0
Topological ends: 1
Entropy: Exact formula: λ = 1.
Gaussian area: Exact formula: F₀,₁ = λ for the normalized shrinker. λ = 1.

Higher-dimensional planes · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Linear subspace ℝⁿ, 3 ≤ n < N
Embeddedness: Embedded: The linear-subspace inclusion is an embedding.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 1
Entropy: Exact formula: λ = 1.
Gaussian area: Exact formula: F₀,₁ = λ for the normalized shrinker. λ = 1.

Nonclosed shrinking curves dense in an annulus

Numerical ODE approximation

Earliest recorded reference: 2010-07-09

Bounded image, complete noncompact domain, nonproper immersion. Gaussian divergence is a direct calculation, not an extra cited entropy theorem.

Irrational turning branch · Euclidean plane ℝ² · Not embedded

Ambient space: Euclidean plane ℝ²
Topology: Noncompact line ℝ
Embeddedness: Not embedded: Repeated rotated excursions intersect; this is a recurrent, nonproper immersion. The annulus is the closure of its image, not its parameter manifold.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2 intrinsic ends of the parameter manifold
Entropy: Exact formula: ∞ at every regular time, with arclength counted on the immersion: infinite length in a compact region makes every Gaussian integral diverge.
Gaussian area: Exact formula: ∞ at every regular time, with arclength counted on the immersion: infinite length in a compact region makes every Gaussian integral diverge.

Cisgeminate-eye shrinker

Source-informed network schematic

Earliest recorded reference: 2019-01-29

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Regular planar network · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Two bounded four-sided cells sharing an edge, with four exterior rays
Embeddedness: Embedded: Embedded as a network with prescribed 120-degree triple junctions, not a smooth curve there.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Four exterior rays.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Embedded rotational shrinking tori (Angenent, 1992; Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017; Berchenko-Kogan, 2018)

Numerical ODE surface

Earliest recorded reference: 1992

Angenent surface torus · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Torus S¹ × S¹
Embeddedness: Embedded: The selected simple closed rotational profile gives an embedded hypersurface in every stated dimension.
Compactness: Compact
Surface genus: 1
Topological ends: 0 (compact)
Entropy: Numerical estimate: λ ≈ 1.85122. Berchenko–Kogan estimate; no certified error enclosure.
Gaussian area: Numerical estimate: F₀,₁ = λ for this normalized shrinker. λ ≈ 1.85122. Berchenko–Kogan estimate; no certified error enclosure.

Higher-dimensional rotational tori · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: S¹ × Sⁿ⁻¹, n ≥ 3
Embeddedness: Embedded: The selected simple closed rotational profile gives an embedded hypersurface in every stated dimension.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite; no value recorded.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for the normalized shrinker. Finite; no value recorded.

Chopp cubical shrinker candidate

Schematic of a numerical candidate

Earliest recorded reference: 1994

Numerical shrinker candidate, not a proved existence result for this computed surface. Topology and compactness describe the intended surface; the drawing is schematic.

Numerical candidate: Chopp cubical shrinker candidate · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Intended orientable genus-5 surface without punctures (closed)
Embeddedness: Not recorded: The target is embedded and the numerical construction uses embedded level sets, but existence of an exact embedded shrinker with these properties is not proved in the reviewed computation.
Compactness: Compact
Surface genus: 5, the topology of the numerical candidate
Topological ends: 0, for the intended complete surface; finite-window boundaries are not ends
Entropy: Not recorded: No entropy value or bound established for this numerical candidate. The self-similarity diagnostic E is not entropy.
Gaussian area: Not recorded: No normalized centered Gaussian area recorded; do not infer it from the drawing or identify it with entropy for an approximate surface.

Chopp octahedral shrinker candidate

Schematic of a numerical candidate

Earliest recorded reference: 1994

Numerical shrinker candidate, not a proved existence result for this computed surface. Topology and compactness describe the intended surface; the drawing is schematic.

Numerical candidate: Chopp octahedral shrinker candidate · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Intended orientable genus-7 surface without punctures (closed)
Embeddedness: Not recorded: The target is embedded and the numerical construction uses embedded level sets, but existence of an exact embedded shrinker with these properties is not proved in the reviewed computation.
Compactness: Compact
Surface genus: 7, the topology of the numerical candidate
Topological ends: 0, for the intended complete surface; finite-window boundaries are not ends
Entropy: Not recorded: No entropy value or bound established for this numerical candidate. The self-similarity diagnostic E is not entropy.
Gaussian area: Not recorded: No normalized centered Gaussian area recorded; do not infer it from the drawing or identify it with entropy for an approximate surface.

Chopp punctured-torus saddle candidate

Schematic of a numerical candidate

Earliest recorded reference: 1994

Numerical shrinker candidate, not a proved existence result for this computed surface. Topology and compactness describe the intended surface; the drawing is schematic.

Numerical candidate: Chopp punctured-torus saddle candidate · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Intended orientable genus-1 surface minus 1 point
Embeddedness: Not recorded: The target is embedded and the numerical construction uses embedded level sets, but existence of an exact embedded shrinker with these properties is not proved in the reviewed computation.
Compactness: Noncompact
Surface genus: 1, the topology of the numerical candidate
Topological ends: 1, for the intended complete surface; finite-window boundaries are not ends
Entropy: Not recorded: No entropy value or bound established for this numerical candidate. The self-similarity diagnostic E is not entropy.
Gaussian area: Not recorded: No normalized centered Gaussian area recorded; do not infer it from the drawing or identify it with entropy for an approximate surface.

Chopp two-handled cone candidate

Schematic of a numerical candidate

Earliest recorded reference: 1994

Numerical shrinker candidate, not a proved existence result for this computed surface. Topology and compactness describe the intended surface; the drawing is schematic.

Numerical candidate: Chopp two-handled cone candidate · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Intended orientable genus-1 surface minus 2 points
Embeddedness: Not recorded: The target is embedded and the numerical construction uses embedded level sets, but existence of an exact embedded shrinker with these properties is not proved in the reviewed computation.
Compactness: Noncompact
Surface genus: 1, the topology of the numerical candidate
Topological ends: 2, for the intended complete surface; finite-window boundaries are not ends
Entropy: Not recorded: No entropy value or bound established for this numerical candidate. The self-similarity diagnostic E is not entropy.
Gaussian area: Not recorded: No normalized centered Gaussian area recorded; do not infer it from the drawing or identify it with entropy for an approximate surface.

Numerical four-handled cone shrinker candidate (Chopp, 1994; Angenent–Ilmanen–Chopp, 1995)

Schematic of a numerical candidate

Earliest recorded reference: 1994

Numerical shrinker candidate, not a proved existence result for this computed surface. Topology and compactness describe the intended surface; the drawing is schematic.

Numerical candidate: Chopp four-handled cone candidate · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Intended orientable genus-3 surface minus 2 points
Embeddedness: Not recorded: The target is embedded and the numerical construction uses embedded level sets, but existence of an exact embedded shrinker with these properties is not proved in the reviewed computation.
Compactness: Noncompact
Surface genus: 3, the topology of the numerical candidate
Topological ends: 2, for the intended complete surface; finite-window boundaries are not ends
Entropy: Not recorded: No entropy value or bound established for this numerical candidate. The self-similarity diagnostic E is not entropy.
Gaussian area: Not recorded: No normalized centered Gaussian area recorded; do not infer it from the drawing or identify it with entropy for an approximate surface.

Generalized shrinking cylinders

Exact formula

Earliest recorded reference: 2009-08-26

Surface cylinder · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Cylinder S¹ × ℝ
Embeddedness: Embedded: The product of the round sphere with its Euclidean factor is embedded.
Compactness: Noncompact
Surface genus: 0
Topological ends: 2
Entropy: Exact formula: λ = √(2π/e) ≈ 1.52035.
Gaussian area: Exact formula: F₀,₁ = λ for the normalized shrinker. λ = √(2π/e) ≈ 1.52035.

Higher-dimensional cylinders with one axial direction · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sⁿ⁻¹ × ℝ, n ≥ 3
Embeddedness: Embedded: The product of the round sphere with its Euclidean factor is embedded.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2
Entropy: Exact formula: λ equals the entropy of the shrinking Sᵏ factor.
Gaussian area: Exact formula: F₀,₁ = λ for the normalized shrinker. λ equals the entropy of the shrinking Sᵏ factor.

Cylinders with at least two Euclidean directions · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sᵏ × ℝⁿ⁻ᵏ, n ≥ 3, 1 ≤ k ≤ n−2
Embeddedness: Embedded: The product of the round sphere with its Euclidean factor is embedded.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 1
Entropy: Exact formula: λ equals the entropy of the shrinking Sᵏ factor.
Gaussian area: Exact formula: F₀,₁ = λ for the normalized shrinker. λ equals the entropy of the shrinking Sᵏ factor.

Rotational shrinking trumpet ends

Numerical ODE end

Earliest recorded reference: 2010-08-10

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Surface end in ℝ³ · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Exterior end or orbit-space disk as specified; boundary retained
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Higher-dimensional ends · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Exterior end or orbit-space disk as specified; boundary retained
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not a surface.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Nguyen's one-ended shrinking saddles

Scherk-based construction schematic

Earliest recorded reference: 2011-06-26

Nguyen uses H-vector = −X-perp. Multiply the source surface by √2 for this catalogue. No identification with the KKM construction is asserted.

Sufficiently large m; genus m−1 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Connected orientable genus-(m−1) surface with one puncture; m sufficiently large.
Embeddedness: Embedded: The constructed one-ended surfaces are smooth and embedded.
Compactness: Noncompact
Surface genus: g=m−1, for every sufficiently large integer m.
Topological ends: One asymptotically conical graphical end.
Entropy: Finite; value not recorded: Finite by quadratic area growth; no scalar value or uniform numerical bound recorded.
Gaussian area: Finite; value not recorded: F₀,₁=λ for M=√2 times the source shrinker; finite, value unrecorded.

One-ended high-genus shrinkers from sphere–plane gluing (Kapouleas–Kleene–Møller, 2011; Sun–Zhichao Wang, 2018)

Scherk-based construction schematic

Earliest recorded reference: 2011-06-27

Existence is proved for every sufficiently large genus, with no explicit threshold. The Scherk-based interactive picture is a construction schematic; its period counts and cone profiles are not computed KKM solutions or entropy data.

Closed genus-g surface minus one point · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Closed genus-g surface minus one point
Embeddedness: Embedded: The constructed surface is embedded after desingularization; the intersecting sphere and plane are its singular limiting configuration.
Compactness: Noncompact
Surface genus: Every sufficiently large genus g
Topological ends: 1 conical end
Entropy: Finite; value not recorded: Finite by the stated compactness or area growth; no value recorded.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for this normalized shrinker. Finite by the stated compactness or area growth; no value recorded.

Closed shrinkers of large even genus via torus–sphere gluing (Colding–Minicozzi II, 2009; Møller, 2011; Drugan–Lee–X. H. Nguyen, 2017)

Numerical limiting configuration

Earliest recorded reference: 2011-11-30

Properties refer to the smooth surfaces in the existence theorem. The figures show the singular sphere–torus limit and illustrative handle models of genera 8, 16, and 24. The latter are topology-checked meshes, not computed shrinkers; the theorem gives no explicit genus threshold, and no entropy is inferred from either picture.

Closed orientable surface · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Closed orientable surface
Embeddedness: Embedded: The constructed finite-genus surfaces are embedded; the intersecting sphere–torus configuration is only their singular limit.
Compactness: Compact
Surface genus: Sufficiently large even genus
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite; no value recorded.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for the normalized shrinker. Finite; no value recorded.

Immersed rotational shrinking spheres (Drugan, 2013; Drugan–Lee–X. H. Nguyen, 2017)

Numerical ODE surface

Earliest recorded reference: 2013-03-29

Topology refers to the parameter domain, not the self-intersecting image.

Immersed two-sphere · Euclidean space ℝ³ · Not embedded

Ambient space: Euclidean space ℝ³
Topology: Immersed sphere S²
Embeddedness: Not embedded: These rotational sphere immersions have self-intersections in every stated dimension.
Compactness: Compact
Surface genus: 0
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite; no value recorded.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for the normalized shrinker. Finite; no value recorded.

Higher-dimensional immersed spheres · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Not embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Immersed sphere Sⁿ, n ≥ 3
Embeddedness: Not embedded: These rotational sphere immersions have self-intersections in every stated dimension.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite; no value recorded.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for the normalized shrinker. Finite; no value recorded.

Drugan–Kleene immersed shrinking families

Source-informed schematic

Earliest recorded reference: 2013-06-10

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Surface cases, n = 2 · Euclidean space ℝ³ · Depends on the construction

Ambient space: Euclidean space ℝ³
Topology: S², ℝ², ℝ × S¹, or S¹ × S¹
Embeddedness: Depends on the construction: The nontrivial sphere, plane and cylinder branches are nonembedded. Torus embeddedness is not asserted uniformly.
Compactness: Not specified for this subfamily
Surface genus: Depends on the construction.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Hypersurface cases, n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Depends on the construction

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sⁿ, ℝⁿ, ℝ × Sⁿ⁻¹, or S¹ × Sⁿ⁻¹
Embeddedness: Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness: Not specified for this subfamily
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Shrinking ends asymptotic to isoparametric cones

Source-informed orbit-space schematic

Earliest recorded reference: 2015-10-24

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Surface end in ℝ³ · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Exterior end or orbit-space disk as specified; boundary retained
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Higher-dimensional ends · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Exterior end or orbit-space disk as specified; boundary retained
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not a surface.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Ketover shrinkers with Platonic symmetry

Interactive construction schematic

Earliest recorded reference: 2016-02-23

Five equivariant min-max constructions. Compactness and ends are not proved; the diagram illustrates sweepout topology. No identification with later constructions is assumed.

Tetrahedral symmetry: genus 3 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Orientable embedded surface of genus 3; T₁₂ rotation symmetry. The doubled 4-face sweepout does not establish compactness of its limit.
Embeddedness: Embedded: All five recorded symmetry/genus constructions give smooth embedded surfaces.
Compactness: Not specified for this subfamily
Surface genus: 3
Topological ends: Not established in the cited existence theorem
Entropy: Proved bound or relation: 4/e < λ < 8/e; exact value not given.
Gaussian area: Proved bound or relation: F₀,₁ = λ for the normalized shrinker; 4/e < F₀,₁ < 8/e.

Cubic symmetry: genus 5 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Orientable embedded surface of genus 5; O₂₄ rotation symmetry. The doubled 6-face sweepout does not establish compactness of its limit.
Embeddedness: Embedded: All five recorded symmetry/genus constructions give smooth embedded surfaces.
Compactness: Not specified for this subfamily
Surface genus: 5
Topological ends: Not established in the cited existence theorem
Entropy: Proved bound or relation: 4/e < λ < 8/e; exact value not given.
Gaussian area: Proved bound or relation: F₀,₁ = λ for the normalized shrinker; 4/e < F₀,₁ < 8/e.

Octahedral symmetry: genus 7 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Orientable embedded surface of genus 7; O₂₄ rotation symmetry. The doubled 8-face sweepout does not establish compactness of its limit.
Embeddedness: Embedded: All five recorded symmetry/genus constructions give smooth embedded surfaces.
Compactness: Not specified for this subfamily
Surface genus: 7
Topological ends: Not established in the cited existence theorem
Entropy: Proved bound or relation: 4/e < λ < 8/e; exact value not given.
Gaussian area: Proved bound or relation: F₀,₁ = λ for the normalized shrinker; 4/e < F₀,₁ < 8/e.

Dodecahedral symmetry: genus 11 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Orientable embedded surface of genus 11; I₆₀ rotation symmetry. The doubled 12-face sweepout does not establish compactness of its limit.
Embeddedness: Embedded: All five recorded symmetry/genus constructions give smooth embedded surfaces.
Compactness: Not specified for this subfamily
Surface genus: 11
Topological ends: Not established in the cited existence theorem
Entropy: Proved bound or relation: 4/e < λ < 8/e; exact value not given.
Gaussian area: Proved bound or relation: F₀,₁ = λ for the normalized shrinker; 4/e < F₀,₁ < 8/e.

Icosahedral symmetry: genus 19 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Orientable embedded surface of genus 19; I₆₀ rotation symmetry. The doubled 20-face sweepout does not establish compactness of its limit.
Embeddedness: Embedded: All five recorded symmetry/genus constructions give smooth embedded surfaces.
Compactness: Not specified for this subfamily
Surface genus: 19
Topological ends: Not established in the cited existence theorem
Entropy: Proved bound or relation: 4/e < λ < 8/e; exact value not given.
Gaussian area: Proved bound or relation: F₀,₁ = λ for the normalized shrinker; 4/e < F₀,₁ < 8/e.

Variational embedded shrinking tori (Drugan–X. H. Nguyen, 2017; Chu–Sun, 2023)

Variational construction schematic

Earliest recorded reference: 2017-08-29

This construction is not proved distinct from, or identical with, Angenent’s shooting torus.

Torus S¹ × S¹ · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Torus S¹ × S¹
Embeddedness: Embedded: The variational construction gives a compact embedded rotational hypersurface in every stated dimension.
Compactness: Compact
Surface genus: 1
Topological ends: 0 (compact)
Entropy: Proved bound or relation: λ < 2 (proved; no individual exact value).
Gaussian area: Proved bound or relation: F₀,₁ = λ for this normalized shrinker. λ < 2 (proved; no individual exact value).

Higher-dimensional variational tori · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: S¹ × Sⁿ⁻¹, n ≥ 3
Embeddedness: Embedded: The variational construction gives a compact embedded rotational hypersurface in every stated dimension.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Proved bound or relation: λ < 2 (proved; no individual exact value).
Gaussian area: Proved bound or relation: F₀,₁ = λ for this normalized shrinker. λ < 2 (proved; no individual exact value).

Lens-shaped shrinking surface clusters

Numerical ODE cluster

Earliest recorded reference: 2018-11-19

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Two-dimensional cluster in ℝ³ · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Two disks and a planar exterior joined along a triple circle
Embeddedness: Embedded: Embedded sheets meeting at the prescribed triple circle; the union is not smooth there.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: One topological end.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Closed shrinkers from spherical doubling

Interactive spherical-doubling construction schematic

Earliest recorded reference: 2020-01-13

The chosen two-latitude subfamily has 2m bridges. The theorem does not provide every small odd genus.

Closed orientable surface from doubling S² · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Closed orientable surface from doubling S²
Embeddedness: Embedded: Each constructed surface is embedded; the doubled sphere occurs only as a limiting configuration.
Compactness: Compact
Surface genus: 2m−1, m sufficiently large
Topological ends: 0 (compact)
Entropy: Proved bound or relation: λ < 8/e ≈ 2.94304 (proved; no individual value).
Gaussian area: Proved bound or relation: F₀,₁ = λ for this normalized shrinker. λ < 8/e ≈ 2.94304 (proved; no individual value).

Sun–Wang–Zhou shrinkers of arbitrarily large entropy

Original schematic

Earliest recorded reference: 2020-11-09

Gaussian-width sequence for dimensions 2–6; a separate Euclidean-product argument extends unbounded entropy with finite index to all dimensions ≥ 2. The theorem does not prescribe genus, ends, or compactness of the base examples.

Gaussian-width surfaces, n = 2 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Connected embedded surface; topology not specified by the width theorem
Embeddedness: Embedded: The Gaussian-width hypersurfaces are smooth and embedded in the stated dimension range; their Euclidean product extensions are also embedded.
Compactness: Not specified for this subfamily
Surface genus: Not established in the cited existence theorem
Topological ends: Not established in the cited existence theorem
Entropy: Proved bound or relation: λ(Σₖ) = ωₖ, with growth comparable to k^(1/(n+1)); finite for each k and unbounded as k → ∞. No individual numerical values given.
Gaussian area: Proved bound or relation: F₀,₁(Σₖ) = λ(Σₖ) = ωₖ for the normalized shrinker; growth comparable to k^(1/(n+1)).

Gaussian-width hypersurfaces, 3 ≤ n ≤ 6 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Connected embedded n-dimensional hypersurface; topology not specified by the width theorem
Embeddedness: Embedded: The Gaussian-width hypersurfaces are smooth and embedded in the stated dimension range; their Euclidean product extensions are also embedded.
Compactness: Not specified for this subfamily
Surface genus: Not a surface
Topological ends: Not established in the cited existence theorem
Entropy: Proved bound or relation: λ(Σₖ) = ωₖ, with growth comparable to k^(1/(n+1)); finite for each k and unbounded as k → ∞. No individual numerical values given.
Gaussian area: Proved bound or relation: F₀,₁(Σₖ) = λ(Σₖ) = ωₖ for the normalized shrinker; growth comparable to k^(1/(n+1)).

Euclidean product extension, n ≥ 7 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Σ⁶ × ℝ^(n−6), for a six-dimensional Gaussian-width shrinker; base topology not specified
Embeddedness: Embedded: The Gaussian-width hypersurfaces are smooth and embedded in the stated dimension range; their Euclidean product extensions are also embedded.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Not established in the cited existence theorem
Entropy: Proved bound or relation: For every Λ > 0 choose λ > Λ finite; λ(Σ⁶ × ℝ^(n−6)) = λ(Σ⁶). Finite Gaussian index is preserved.
Gaussian area: Proved bound or relation: F₀,₁ = λ for the normalized product shrinker; equal to the Gaussian area of the base.

Noncompact shrinkers of every positive genus (Buzano–H. T. Nguyen–Schulz, 2021; Chu–Sun, 2023)

Interactive topology schematic

Earliest recorded reference: 2021-10-12

The one-end theorem is restricted to sufficiently large genus. Coincidence with the KKM construction is not proved.

Genus one · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Noncompact genus-one surface with finitely many annular ends
Embeddedness: Embedded: The Buzano–Nguyen–Schulz construction gives an embedded surface for every positive genus, including all stated subfamilies.
Compactness: Noncompact
Surface genus: 1
Topological ends: At least one conical end; total number not established here
Entropy: Proved bound or relation: 1 < λ ≤ 1.02 + 4/e < 3. The genus-one simulation 1.7143 is not a certified value for the constructed family.
Gaussian area: Proved bound or relation: F₀,₁ = λ for this normalized shrinker. 1 < λ ≤ 1.02 + 4/e < 3. The genus-one simulation 1.7143 is not a certified value for the constructed family.

Arbitrary positive genus · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Noncompact genus-g surface with finitely many annular ends
Embeddedness: Embedded: The Buzano–Nguyen–Schulz construction gives an embedded surface for every positive genus, including all stated subfamilies.
Compactness: Noncompact
Surface genus: Every g ≥ 2
Topological ends: At least one conical end; total number not established for every g
Entropy: Proved bound or relation: 1 < λ ≤ 1.02 + 4/e < 3. The genus-one simulation 1.7143 is not a certified value for the constructed family.
Gaussian area: Proved bound or relation: F₀,₁ = λ for this normalized shrinker. 1 < λ ≤ 1.02 + 4/e < 3. The genus-one simulation 1.7143 is not a certified value for the constructed family.

Sufficiently large genus · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Closed genus-g surface minus one point
Embeddedness: Embedded: The Buzano–Nguyen–Schulz construction gives an embedded surface for every positive genus, including all stated subfamilies.
Compactness: Noncompact
Surface genus: Sufficiently large genus g
Topological ends: Exactly 1 conical end
Entropy: Proved bound or relation: 1 < λ ≤ 1.02 + 4/e < 3. The genus-one simulation 1.7143 is not a certified value for the constructed family.
Gaussian area: Proved bound or relation: F₀,₁ = λ for this normalized shrinker. 1 < λ ≤ 1.02 + 4/e < 3. The genus-one simulation 1.7143 is not a certified value for the constructed family.

Chu–Sun genus-one shrinkers below the shrinking doughnut

Original schematic

Earliest recorded reference: 2023-08-11

A multiplicity-one genus-one tangent-flow model with entropy strictly below the Drugan–Nguyen shrinking doughnut. Compactness and ends are not specified. No identification with the BNS genus-one shrinker is established.

Lower-entropy genus-one shrinker · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Embedded orientable genus-one surface; compactness not specified
Embeddedness: Embedded: The lower-entropy genus-one shrinker supplied by the cited existence theorem is embedded.
Compactness: Not specified for this subfamily
Surface genus: 1
Topological ends: Not established in the cited existence theorem
Entropy: Proved bound or relation: λ(Σ) < λ(Drugan–Nguyen torus) < 2; no exact value supplied.
Gaussian area: Proved bound or relation: F₀,₁(Σ) = λ(Σ) < λ(Drugan–Nguyen torus) < 2.

Ilmanen–White two-ended shrinkers

Interactive topology schematic

Earliest recorded reference: 2024-06-26

Write h for shrinker genus; the source uses a compact wheel of genus g=h+1. This construction is not identified with Ketover’s family.

All genera h ≥ 2 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Closed orientable genus-h surface with two punctures
Embeddedness: Embedded: The constructed two-ended surfaces are properly embedded for every stated genus.
Compactness: Noncompact
Surface genus: Every integer h ≥ 2
Topological ends: Exactly 2, both asymptotically conical
Entropy: Finite; value not recorded: Finite; an exact value or numerical upper bound is not recorded for this construction.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for the normalized shrinker. Finite; an exact value or numerical upper bound is not recorded for this construction.

Ketover self-shrinking doubled planes

Interactive doubled-plane schematic

Earliest recorded reference: 2024-07-01

Compactness and end count are not inferred for the small-genus members. Entropy tends to 2 as g tends to infinity.

Genus one · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Embedded genus-one surface
Embeddedness: Embedded: The surfaces are properly embedded for every stated genus. A multiplicity-two plane occurs only in the large-genus limit.
Compactness: Not specified for this subfamily
Surface genus: 1
Topological ends: Not established for this member
Entropy: Proved bound or relation: λ < 2 (proved; no individual exact value).
Gaussian area: Proved bound or relation: F₀,₁ = λ for this normalized shrinker. λ < 2 (proved; no individual exact value).

Arbitrary positive genus · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Embedded genus-g surface
Embeddedness: Embedded: The surfaces are properly embedded for every stated genus. A multiplicity-two plane occurs only in the large-genus limit.
Compactness: Not specified for this subfamily
Surface genus: Every g ≥ 2
Topological ends: Not established for every g
Entropy: Proved bound or relation: λ < 2 (proved; no individual exact value).
Gaussian area: Proved bound or relation: F₀,₁ = λ for this normalized shrinker. λ < 2 (proved; no individual exact value).

Sufficiently large genus · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Closed genus-g surface minus two points
Embeddedness: Embedded: The surfaces are properly embedded for every stated genus. A multiplicity-two plane occurs only in the large-genus limit.
Compactness: Noncompact
Surface genus: Sufficiently large genus g
Topological ends: Exactly 2 conical ends
Entropy: Proved bound or relation: λ < 2 (proved; no individual exact value).
Gaussian area: Proved bound or relation: F₀,₁ = λ for this normalized shrinker. λ < 2 (proved; no individual exact value).

Flow-generated Platonic and polygonal shrinkers

Original construction schematic

Earliest recorded reference: 2025-02-27

MCF construction, separate from the min-max entries; no equality or noncongruence theorem is asserted. Compact numerical pictures do not determine compactness of the theorem’s selected surfaces. The interactive punctured-shell models illustrate the topology of the initial-surface construction in Theorem 24; their compactness and display parameters are not properties of the tangent shrinkers.

Platonic face-center ray configurations · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Orientable surface of genus q−1, where q is the number of faces; compactness and ends unresolved
Embeddedness: Embedded: The flow-generated shrinkers are smooth and embedded in all recorded symmetry subfamilies.
Compactness: Not specified for this subfamily
Surface genus: 3, 5, 7, 11, 19 for tetrahedron, cube, octahedron, dodecahedron, icosahedron
Topological ends: Not determined by Theorem 12
Entropy: Finite; value not recorded: Finite; an exact value or numerical upper bound is not recorded for this construction.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for the normalized shrinker. Finite; an exact value or numerical upper bound is not recorded for this construction.

Regular n-gon ray configurations, n ≥ 3 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Orientable surface of genus n−1; compactness and end count not specified in the general theorem
Embeddedness: Embedded: The flow-generated shrinkers are smooth and embedded in all recorded symmetry subfamilies.
Compactness: Not specified for this subfamily
Surface genus: n−1, n ≥ 3
Topological ends: Not determined by Theorem 12
Entropy: Finite; value not recorded: Finite; an exact value or numerical upper bound is not recorded for this construction.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for the normalized shrinker. Finite; an exact value or numerical upper bound is not recorded for this construction.

Regular n-gon, sufficiently large n · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Noncompact orientable genus-(n−1) surface
Embeddedness: Embedded: The flow-generated shrinkers are smooth and embedded in all recorded symmetry subfamilies.
Compactness: Noncompact
Surface genus: n−1 for sufficiently large n
Topological ends: Exact end count not recorded
Entropy: Finite; value not recorded: Finite; an exact value or numerical upper bound is not recorded for this construction.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for the normalized shrinker. Finite; an exact value or numerical upper bound is not recorded for this construction.

Flow-generated sphere–plane shrinkers

Topology schematic and limiting configuration

Earliest recorded reference: 2025-02-27

The Euclidean shrinker is a tangent flow of an intrinsic compact flow in S²×ℝ. Choose 1+4/e < δ < 1+√(2π/e); the constructed member depends on this choice.

Genus-one member · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Noncompact orientable genus-one surface containing two horizontal lines
Embeddedness: Embedded: The flow-generated surfaces are properly embedded for every stated genus, including the genus-one member.
Compactness: Noncompact
Surface genus: 1
Topological ends: Exact number not specified for small genus
Entropy: Proved bound or relation: λ < δ < 1+√(2π/e) < 3, for a chosen δ > 1+4/e.
Gaussian area: Proved bound or relation: F₀,₁ = λ for the normalized shrinker. λ < δ < 1+√(2π/e) < 3, for a chosen δ > 1+4/e.

Genus g ≥ 2 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Noncompact orientable genus-g surface containing g+1 horizontal lines
Embeddedness: Embedded: The flow-generated surfaces are properly embedded for every stated genus, including the genus-one member.
Compactness: Noncompact
Surface genus: Every integer g ≥ 2
Topological ends: Exact number not specified in the all-genus theorem
Entropy: Proved bound or relation: λ < δ < 1+√(2π/e) < 3, for a chosen δ > 1+4/e.
Gaussian area: Proved bound or relation: F₀,₁ = λ for the normalized shrinker. λ < δ < 1+√(2π/e) < 3, for a chosen δ > 1+4/e.

Sufficiently large genus · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Closed orientable genus-g surface with one puncture
Embeddedness: Embedded: The flow-generated surfaces are properly embedded for every stated genus, including the genus-one member.
Compactness: Noncompact
Surface genus: Every sufficiently large integer g
Topological ends: Exactly 1 asymptotically conical end
Entropy: Proved bound or relation: λ < δ < 1+√(2π/e) < 3, for a chosen δ > 1+4/e.
Gaussian area: Proved bound or relation: F₀,₁ = λ for the normalized shrinker. λ < δ < 1+√(2π/e) < 3, for a chosen δ > 1+4/e.

Flow-generated three-ended shrinkers

Original topology schematic

Earliest recorded reference: 2025-02-27

The genus 2g and exactly three ends are proved only for g sufficiently large. The Euclidean shrinker arises from an initial compact flow in S²×ℝ.

Large-genus three-ended subfamily · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Closed orientable genus-2g surface with three punctures
Embeddedness: Embedded: The constructed three-ended surfaces are smooth and embedded.
Compactness: Noncompact
Surface genus: 2g for sufficiently large integers g
Topological ends: Exactly 3 asymptotically conical annular ends
Entropy: Proved bound or relation: λ ≤ Λ < ∞ uniformly in g; no numerical value for Λ is specified.
Gaussian area: Proved bound or relation: F₀,₁ = λ for the normalized shrinker. λ ≤ Λ < ∞ uniformly in g; no numerical value for Λ is specified.

Shao–Zou shrinkers with any prescribed number of ends

Interactive construction schematic

Earliest recorded reference: 2025-07-24

For each fixed k ≥ 2, m must be sufficiently large depending on k. There are m bridges between each pair of neighboring sheets.

Two stacked planes · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Closed orientable genus-(m−1) surface with two punctures
Embeddedness: Embedded: Each finite-parameter surface is embedded with multiplicity one. The stacked planar limit is not a member of the smooth family.
Compactness: Noncompact
Surface genus: m−1 for sufficiently large m
Topological ends: Exactly 2 conical ends
Entropy: Finite; value not recorded: Finite; an exact value or numerical upper bound is not recorded for this construction.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for the normalized shrinker. Finite; an exact value or numerical upper bound is not recorded for this construction.

Three stacked planes · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Closed orientable genus-2(m−1) surface with three punctures
Embeddedness: Embedded: Each finite-parameter surface is embedded with multiplicity one. The stacked planar limit is not a member of the smooth family.
Compactness: Noncompact
Surface genus: 2(m−1) for sufficiently large m
Topological ends: Exactly 3 conical ends
Entropy: Finite; value not recorded: Finite; an exact value or numerical upper bound is not recorded for this construction.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for the normalized shrinker. Finite; an exact value or numerical upper bound is not recorded for this construction.

k stacked planes, k ≥ 4 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Closed orientable genus-(k−1)(m−1) surface with k punctures
Embeddedness: Embedded: Each finite-parameter surface is embedded with multiplicity one. The stacked planar limit is not a member of the smooth family.
Compactness: Noncompact
Surface genus: (k−1)(m−1), m sufficiently large for fixed k
Topological ends: Exactly k conical ends, any fixed k ≥ 4
Entropy: Finite; value not recorded: Finite; an exact value or numerical upper bound is not recorded for this construction.
Gaussian area: Finite; value not recorded: F₀,₁ = λ for the normalized shrinker. Finite; an exact value or numerical upper bound is not recorded for this construction.

Products of closed shrinking curves

Factor diagram

Earliest recorded reference: 2005-07-15

The all-circle m=2 case is also listed as the Clifford torus. Topology refers to the immersion domain.

Two curve factors · Euclidean space ℝ⁴ · Depends on the construction

Ambient space: Euclidean space ℝ⁴
Topology: Immersed torus (S¹)²
Embeddedness: Depends on the construction: Embedded exactly when every factor is an embedded, singly covered circle. A noncircular Abresch–Langer factor gives self-intersections; a multiple cover is not an embedding.
Compactness: Compact
Surface genus: 1
Topological ends: 0 (compact)
Entropy: Exact formula: λ = ∏ⱼ λ(γⱼ), with immersion multiplicity counted; depends on the chosen factors.
Gaussian area: Exact formula: F₀,₁ = λ for this normalized shrinker. λ = ∏ⱼ λ(γⱼ), with immersion multiplicity counted; depends on the chosen factors.

Three or more curve factors · Euclidean dimension ≥ 5 · Depends on the construction

Ambient space: Euclidean dimension ≥ 5
Topology: Immersed m-torus, m ≥ 3, in ℝ²ᵐ
Embeddedness: Depends on the construction: Embedded exactly when every factor is an embedded, singly covered circle. A noncircular Abresch–Langer factor gives self-intersections; a multiple cover is not an embedding.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Exact formula: λ = ∏ⱼ λ(γⱼ), with immersion multiplicity counted; depends on the chosen factors.
Gaussian area: Exact formula: F₀,₁ = λ for this normalized shrinker. λ = ∏ⱼ λ(γⱼ), with immersion multiplicity counted; depends on the chosen factors.

Higher-codimension spherical shrinkers (Smoczyk, 2005; Sun–Zhu, 2020)

Dimension and factor diagram

Earliest recorded reference: 2005-07-15

Topology depends on the chosen spherical minimal immersion; the explicit product subfamily has topology Sᵖ × Sᑫ.

Surface inputs · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Depends on the construction

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Compact minimal surface in a sphere
Embeddedness: Depends on the construction: Embedded exactly when the chosen minimal immersion into the sphere is embedded. The explicit products of round spheres are embedded; general immersed inputs need not be.
Compactness: Compact
Surface genus: Depends on parameters
Topological ends: 0 (compact)
Entropy: Exact formula: λ = (4π)^(−n/2) e^(−n/2) Area(M). Spherical products have λ = λ(Sᵖ) λ(Sᑫ).
Gaussian area: Exact formula: F₀,₁ = λ for this normalized shrinker. λ = (4π)^(−n/2) e^(−n/2) Area(M). Spherical products have λ = λ(Sᵖ) λ(Sᑫ).

Higher-dimensional inputs · Euclidean dimension ≥ 5 · Depends on the construction

Ambient space: Euclidean dimension ≥ 5
Topology: Compact minimal n-manifold in a sphere; n ≥ 3
Embeddedness: Depends on the construction: Embedded exactly when the chosen minimal immersion into the sphere is embedded. The explicit products of round spheres are embedded; general immersed inputs need not be.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Exact formula: λ = (4π)^(−n/2) e^(−n/2) Area(M). Spherical products have λ = λ(Sᵖ) λ(Sᑫ).
Gaussian area: Exact formula: F₀,₁ = λ for this normalized shrinker. λ = (4π)^(−n/2) e^(−n/2) Area(M). Spherical products have λ = λ(Sᵖ) λ(Sᑫ).

An embedded self-shrinking Möbius bundle (Lee–Mu-Tao Wang, 2007; Braxton–Lee–Zhu, 2026)

Exact four-dimensional projection

Earliest recorded reference: 2007-07-02

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Single-covered Möbius bundle in C² · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: Open Möbius line bundle; one crosscap
Embeddedness: Embedded: Embedded after (θ + π, −r) ~ (θ, r). The cylindrical parametrization before quotient is a nonembedding double cover.
Compactness: Noncompact
Surface genus: Nonorientable surface with one crosscap; orientable genus is inapplicable.
Topological ends: One end; the two r-directions are identified by the quotient.
Entropy: Exact formula: λ = (√2 e^(1/8)/8)[K₀(3/8) + 3K₁(3/8)], for the embedded quotient counted once. Kⱼ is the modified Bessel function of the second kind. The formula is exact; no rounded scalar is used in proved-bound filtering.
Gaussian area: Exact formula: F₀,₁ = λ for this normalized proper asymptotically conical shrinker. λ = (√2 e^(1/8)/8)[K₀(3/8) + 3K₁(3/8)], for the embedded quotient counted once. Kⱼ is the modified Bessel function of the second kind. The formula is exact; no rounded scalar is used in proved-bound filtering.

Compact periodic Lagrangian quadric shrinkers

Construction diagram

Earliest recorded reference: 2008-01-24

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Nonconstant periodic quadric shrinker in C² · Euclidean space ℝ⁴ · Not recorded

Ambient space: Euclidean space ℝ⁴
Topology: Torus S¹ × S¹ (immersion domain)
Embeddedness: Not recorded: The theorem establishes an immersion of Sⁿ⁻¹ × S¹. It does not establish embeddedness for every closing datum; the constant-radius subcase is excluded.
Compactness: Compact
Surface genus: 1 for the torus parameter domain in C².
Topological ends: No ends (compact domain).
Entropy: Finite; value not recorded: Compact smooth shrinker immersion, with multiplicity counted: F₀,₁ = λ is finite in the normalization H = −X⊥/2; no value recorded.
Gaussian area: Finite; value not recorded: Compact smooth shrinker immersion, with multiplicity counted: F₀,₁ = λ is finite in the normalization H = −X⊥/2; no value recorded.

Nonconstant periodic quadric shrinker in Cⁿ, n ≥ 3 · Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean dimension ≥ 5
Topology: Sⁿ⁻¹ × S¹
Embeddedness: Not recorded: The theorem establishes an immersion of Sⁿ⁻¹ × S¹. It does not establish embeddedness for every closing datum; the constant-radius subcase is excluded.
Compactness: Compact
Surface genus: Intrinsic dimension n ≥ 3; surface genus does not apply.
Topological ends: No ends (compact domain).
Entropy: Finite; value not recorded: Compact smooth shrinker immersion, with multiplicity counted: F₀,₁ = λ is finite in the normalization H = −X⊥/2; no value recorded.
Gaussian area: Finite; value not recorded: Compact smooth shrinker immersion, with multiplicity counted: F₀,₁ = λ is finite in the normalization H = −X⊥/2; no value recorded.

Noncompact periodic Lagrangian shrinkers

Source-informed schematic

Earliest recorded reference: 2008-01-24

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

n = 2, m = 1 · Euclidean space ℝ⁴ · Not recorded

Ambient space: Euclidean space ℝ⁴
Topology: Periodic parameter domain S¹ × S⁰ × ℝ; image identifications may occur
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Genus zero.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

n ≥ 3 · Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean dimension ≥ 5
Topology: S¹ × Sᵐ⁻¹ × ℝⁿ⁻ᵐ, as parameter domain
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

The Clifford shrinking torus in four-space

Exact R4 geometry, projected to R3

Earliest recorded reference: 2018-02-05

Torus S¹ × S¹ · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: Torus S¹ × S¹
Embeddedness: Embedded: The product of two singly covered circles is embedded in ℝ⁴. Crossings in a three-dimensional projection are not self-intersections of this torus.
Compactness: Compact
Surface genus: 1
Topological ends: 0 (compact)
Entropy: Exact formula: λ = 2π/e ≈ 2.31145.
Gaussian area: Exact formula: F₀,₁ = λ for the normalized shrinker. λ = 2π/e ≈ 2.31145.

Cylinders over embedded rotational shrinking tori (Angenent, 1992; Berchenko-Kogan, 2018; Chen–Lee–Sun–Zhao, 2026)

Numerical R4 product slices

Earliest recorded reference: 2026-07-26

The product formula and entropy factorization are exact; the torus factor and its quoted entropy are numerical in the picture. Surface genus does not apply to the product.

Torus times one Euclidean direction · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: T²×R
Embeddedness: Embedded: The product of the embedded Angenent torus with a Euclidean factor is embedded.
Compactness: Noncompact
Surface genus: Not a surface: intrinsic dimension k+2≥3; its torus factor has genus one.
Topological ends: Two ends with compact torus cross-sections.
Entropy: Numerical estimate: Exactly λ(T²×Rᵏ)=λ(T²); the torus estimate ≈1.85122 is numerical, without certified error bounds.
Gaussian area: Numerical estimate: Exactly F₀,₁(T²×Rᵏ)=F₀,₁(T²)=λ(T²); numerical estimate ≈1.85122.

Torus times at least two Euclidean directions · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: T²×Rᵏ, k≥2
Embeddedness: Embedded: The product of the embedded Angenent torus with a Euclidean factor is embedded.
Compactness: Noncompact
Surface genus: Not a surface: intrinsic dimension k+2≥3; its torus factor has genus one.
Topological ends: One topological end for k≥2.
Entropy: Numerical estimate: Exactly λ(T²×Rᵏ)=λ(T²); the torus estimate ≈1.85122 is numerical, without certified error bounds.
Gaussian area: Numerical estimate: Exactly F₀,₁(T²×Rᵏ)=F₀,₁(T²)=λ(T²); numerical estimate ≈1.85122.

Park bi-rotational shrinking ends

Source-informed orbit-space schematic

Earliest recorded reference: 2026-08-11

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Boundary construction in the stated dimensions · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Exterior end or orbit-space disk as specified; boundary retained
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Not specified for this subfamily
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Complete bi-rotational shrinking continuations

Numerical ODE segment

Earliest recorded reference: 2026-10-19

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Maximal continued profiles, p,q ≥ 1 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Depends on axis contacts, periodic closure and recurrent behavior
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Not specified for this subfamily
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Lagrangian shrinkers with minimal Legendrian leaves (Anciaux, 2006; Madnick–Wood, 2022)

Numerical ODE profile and projected immersion

Earliest recorded reference: 2006

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Non-spherical compact Legendrian leaves, n ≥ 3 · Euclidean dimension ≥ 5 · Depends on the construction

Ambient space: Euclidean dimension ≥ 5
Topology: S¹ × M (parameter domain; possible image quotients)
Embeddedness: Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Lee–Wang embedded quadric shrinkers

Exact time slices and projected circle action

Earliest recorded reference: 2008-02-04

Full Euclidean ambient dimension is 2n, at least six. The time viewer uses n = 3, k = 2: two-dimensional phase leaves and a projection of sampled circle orbits, not a surface flow in R³. Projection crossings do not change embeddedness. The optional t ≥ 0 cone and expander illustrate the source’s Brakke continuation; properties here concern smooth t < 0 shrinker slices.

Embedded unit-weight shrinkers in Cⁿ, n ≥ 3, n/2 < k < n · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Mapping torus of Sᵏ⁻¹ × Rⁿ⁻ᵏ under the antipodal gluing; not in general a product.
Embeddedness: Embedded: Embedded for this explicitly restricted case; no claim about other weights, groups, or singular limits.
Compactness: Noncompact
Surface genus: Intrinsic dimension at least three; surface genus is inapplicable.
Topological ends: One end, including when the gluing exchanges two fiber ends.
Entropy: Finite; value not recorded: F₀,₁ = λ < ∞ for the complete proper normalized shrinker with polynomial volume growth; no value recorded.
Gaussian area: Finite; value not recorded: F₀,₁ = λ < ∞ for the complete proper normalized shrinker with polynomial volume growth; no value recorded.

Riedler shrinkers from isoparametric foliations

Source-informed schematic

Earliest recorded reference: 2022-07-11

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

g = 3, 4 or 6, with equal admissible multiplicity · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: S¹ × M; n−1 = gm; ambient ℝⁿ⁺¹, starting in ℝ⁵
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Yamamoto Lagrangian shrinkers in Calabi–Yau cones

Source-informed schematic

Earliest recorded reference: 2012-03-18

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Threefold in the six-dimensional cone M_g · Riemannian ambient manifold · Not recorded

Ambient space: Riemannian ambient manifold
Topology: Σ_g × S¹, g ≥ 1
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

The grim reaper and its cylinders (Hoffman–Ilmanen–Martín–White, 2018; Hoffman–Ilmanen–Martín–White, 2019)

Exact formula

Earliest recorded reference: 2018-05-28

The two vertical asymptotes of the generating curve are not two topological ends of the surface cylinder.

Grim-reaper curve · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Line ℝ
Embeddedness: Embedded: The grim-reaper graph and its Euclidean products are properly embedded.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Grim-reaper cylinder · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ²
Embeddedness: Embedded: The grim-reaper graph and its Euclidean products are properly embedded.
Compactness: Noncompact
Surface genus: 0
Topological ends: 1
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Higher-dimensional products · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Euclidean space ℝⁿ, n ≥ 3
Embeddedness: Embedded: The grim-reaper graph and its Euclidean products are properly embedded.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 1
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

The rotational bowl translator (Clutterbuck–Schnürer–Schulze, 2005; Brendle–Choi, 2017; Hoffman–Ilmanen–Martín–White, 2019; Sun–Zhihan Wang, 2023)

Numerical ODE surface

Earliest recorded reference: 2005-09-16

Surface bowl · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ²
Embeddedness: Embedded: The entire rotational graph is properly embedded.
Compactness: Noncompact
Surface genus: 0
Topological ends: 1
Entropy: Exact formula: λ = √(2π/e) ≈ 1.52035.
Gaussian area: Not recorded: Not recorded in this entry.

Higher-dimensional bowls · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Euclidean space ℝⁿ, n ≥ 3
Embeddedness: Embedded: The entire rotational graph is properly embedded.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 1
Entropy: Exact formula: λ equals the entropy of Sⁿ⁻¹(√(2(n−1))); depends on n.
Gaussian area: Not recorded: Not recorded in this entry.

Rotational translating catenoids (Clutterbuck–Schnürer–Schulze, 2005; Hoffman–Ilmanen–Martín–White, 2019)

Numerical ODE surface

Earliest recorded reference: 2005-09-16

Annulus S¹ × ℝ · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Annulus S¹ × ℝ
Embeddedness: Embedded: The two graphical pieces join at a smooth neck to form an embedded hypersurface.
Compactness: Noncompact
Surface genus: 0
Topological ends: 2
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Higher-dimensional winglike translators · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sⁿ⁻¹ × ℝ, n ≥ 3
Embeddedness: Embedded: The two graphical pieces join at a smooth neck to form an embedded hypersurface.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Periodic translating tridents (X. H. Nguyen, 2009; Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019)

Construction schematic

Earliest recorded reference: 2009

The picture is a construction schematic. Its neck scale and limiting half-width do not specify a computed translator. One lower and three upper limiting planes are height-direction asymptotics, not a topological end count.

Periodic surface of infinite genus · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Periodic surface of infinite genus
Embeddedness: Embedded: Complete and properly embedded periodic tridents; Nguyen’s small-scale examples coincide with the semigraphical family by Hoffman–Martín–White, Theorem 1 and p. 2. The intersecting plane and grim cylinder are limiting inputs.
Compactness: Noncompact
Surface genus: Infinite genus
Topological ends: Not recorded
Entropy: Exact formula: λ = 3; Hoffman–Martín–White, arXiv:1909.09241v2, p. 3. Not inferred from the drawing or equated with F₀,₁.
Gaussian area: Not recorded: Not recorded in this entry.

Translators from grim-reaper configurations (X. H. Nguyen, 2010; X. H. Nguyen, 2012)

Source-informed schematic

Earliest recorded reference: 2010-04-15

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Finite configuration: singly periodic surface · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Configuration-dependent periodic surface
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Periodic configuration: doubly periodic surface · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Configuration-dependent doubly periodic surface
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Halldorsson helicoidal rotating translators

Numerical ODE approximation

Earliest recorded reference: 2011-06-22

Finite pitch h > 0; rotation at unit speed equals translation with velocity −h e₃ as unparametrized images. Two profile arms do not give two surface ends.

Symmetric profile through the axis (d = 0, h > 0) · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ²
Embeddedness: Embedded: Central symmetry and strict radial monotonicity of the two arms give an embedded generating curve, hence an embedded surface; see Figure 1 and Section 4.
Compactness: Noncompact
Surface genus: 0
Topological ends: One end of the parameter plane
Entropy: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.
Gaussian area: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.

Profiles with positive minimum radius (d > 0, h > 0) · Euclidean space ℝ³ · Depends on the construction

Ambient space: Euclidean space ℝ³
Topology: Parameter plane ℝ²
Embeddedness: Depends on the construction: The surface is embedded exactly when the two individually embedded profile arms are disjoint. Crossings occur in this family; the source gives no complete threshold in h and d.
Compactness: Noncompact
Surface genus: 0
Topological ends: One end of the parameter plane
Entropy: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.
Gaussian area: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.

Finite-genus translators with three ends (Dávila–del Pino–X. H. Nguyen, 2015; Hoffman–Ilmanen–Martín–White, 2019)

Construction schematic

Earliest recorded reference: 2015-01-16

Orientable surface with three ends · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Orientable surface with three ends
Embeddedness: Embedded: The constructed finite-genus translators are embedded; the intersecting bowl and translating catenoid are construction inputs.
Compactness: Noncompact
Surface genus: Sufficiently large genus
Topological ends: 3
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Tilted grim-reaper cylinders

Exact formula

Earliest recorded reference: 2018-05-28

Plane ℝ² · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ²
Embeddedness: Embedded: The tilted graph and its stated Euclidean products are properly embedded.
Compactness: Noncompact
Surface genus: 0
Topological ends: 1
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Products in higher dimensions · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Euclidean products, topology ℝⁿ
Embeddedness: Embedded: The tilted graph and its stated Euclidean products are properly embedded.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 1
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Strictly convex delta-wing translators

Numerical PDE surface

Earliest recorded reference: 2018-05-28

The two limiting grim cylinders describe directional asymptotics, not separate topological ends.

Plane ℝ² (graph over a strip) · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ² (graph over a strip)
Embeddedness: Embedded: The translating graph over the open strip is embedded.
Compactness: Noncompact
Surface genus: 0
Topological ends: 1
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Translators with prescribed apex curvatures

Numerical solutions and exact formula

Earliest recorded reference: 2018-05-28

Surface cases · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ² (entire or strip graph)
Embeddedness: Embedded: The construction gives properly embedded translating graphs over their stated domains.
Compactness: Noncompact
Surface genus: 0
Topological ends: 1
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Higher-dimensional graphs · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Entire or slab graph, topology ℝⁿ
Embeddedness: Embedded: The construction gives properly embedded translating graphs over their stated domains.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 1
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Scherkenoid translators (Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019)

Connected construction schematic

Earliest recorded reference: 2019-01-25

The source family has 0 < α < π and w ≥ π at unit translation speed. The illustration fixes α = π/2 and varies w. Asymptotic planes in a height direction are not automatically topological ends; the displayed joining shape is schematic.

Periodic surface; global topology not recorded · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Periodic surface; global topology not recorded
Embeddedness: Embedded: The complete Scherkenoid translators in the cited construction are properly embedded.
Compactness: Noncompact
Surface genus: Not recorded
Topological ends: Not recorded
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

The pitchfork translator (Hoffman–Ilmanen–Martín–White, 2019; Hoffman–Martín–White, 2019; Gama–Martín–Møller, 2022; Martín–Sáez–Tsiamis–White, 2024)

Source-informed schematic

Earliest recorded reference: 2019-01-25

The one-plane/three-plane asymptotics do not give four topological ends.

Plane ℝ² (simply connected noncompact surface) · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ² (simply connected noncompact surface)
Embeddedness: Embedded: The complete pitchfork translator is properly embedded.
Compactness: Noncompact
Surface genus: 0
Topological ends: 1
Entropy: Exact formula: λ = 3; Gama–Martín–Møller, Theorem 1.4 and the pitchfork asymptotics.
Gaussian area: Not recorded: Not recorded in this entry.

Doubly periodic Scherk translators (Hoffman–Martín–White, 2019; Martín–Sáez–Tsiamis–White, 2024)

Numerical PDE approximation with illustrative corner fills

Earliest recorded reference: 2019-03-11

Scherk translator in three-space · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Doubly periodic surface of infinite genus
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: Infinite genus
Topological ends: Global topological ends not recorded
Entropy: Exact formula: λ = +∞; infinitely many asymptotic planes (Hoffman–Martín–White, p. 9).
Gaussian area: Not recorded: No fixed-center Gaussian-area value recorded; it is not identified with entropy.

Helicoid-like translators (Hoffman–Martín–White, 2019; Martín–Sáez–Tsiamis–White, 2024)

Heuristic schematic

Earliest recorded reference: 2019-03-11

Helicoid-like surface · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ² (from the strip decomposition)
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: 0
Topological ends: 1
Entropy: Exact formula: λ = +∞; the source gives unbounded area ratios.
Gaussian area: Not recorded: No fixed-center Gaussian-area value recorded; it is not identified with entropy.

Annuloid translators in slabs (Gama–Martín–Møller, 2022; Hoffman–Martín–White, 2023)

Heuristic schematic

Earliest recorded reference: 2023-08-04

Annuloid · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Annulus S¹ × ℝ
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: 0
Topological ends: 2, despite four vertical asymptotic planes
Entropy: Exact formula: λ = 4 (Hoffman–Martín–White, Remark 7.9).
Gaussian area: Not recorded: No fixed-center Gaussian-area value recorded; it is not identified with entropy.

Nonrotational entire convex translators

Level-surface schematic and exact limit model

Earliest recorded reference: 2004-04-19

The picture shows a schematic level surface in the base ℝ³ of an n = 3 graph in ℝ⁴; its surface topology is not the topology of the translator. Section 5 retains rotation symmetry in x′ and reflection in xₙ. Wang’s k-rotational symmetry also requires independence of the remaining coordinates.

Euclidean space ℝⁿ, n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Euclidean space ℝⁿ, n ≥ 3
Embeddedness: Embedded: The smooth entire translating graphs are embedded.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 1
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Joyce–Lee–Tsui small-angle Lagrangian translators

Numerical quadrature and projection

Earliest recorded reference: 2008-01-24

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Lagrangian translator in C² · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: R²
Embeddedness: Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness: Noncompact
Surface genus: 0 (plane topology).
Topological ends: One end of Rⁿ, n ≥ 2.
Entropy: Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Gaussian area: Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.

Lagrangian translator in Cⁿ, n ≥ 3 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Rⁿ
Embeddedness: Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness: Noncompact
Surface genus: Intrinsic dimension n ≥ 3; surface genus does not apply.
Topological ends: One end of Rⁿ, n ≥ 2.
Entropy: Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Gaussian area: Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.

Explicit rotating Lagrangian translators

Exact four-dimensional projection

Earliest recorded reference: 2008-01-24

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Lagrangian translator in C² · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: R²
Embeddedness: Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness: Noncompact
Surface genus: 0 (plane topology).
Topological ends: One end of Rⁿ, n ≥ 2.
Entropy: Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Gaussian area: Finite; value not recorded: The exact parametrization has area density 1+x² and |X|²=x²+x⁴/4+s², so the centered Gaussian integral is finite. This does not assert finite entropy.

Lagrangian translator in Cⁿ, n ≥ 3 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Rⁿ
Embeddedness: Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness: Noncompact
Surface genus: Intrinsic dimension n ≥ 3; surface genus does not apply.
Topological ends: One end of Rⁿ, n ≥ 2.
Entropy: Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Gaussian area: Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.

Castro–Lerma Lagrangian translators from curve pairs

Numerical ODE and integral parametrization

Earliest recorded reference: 2010-07-12

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Complete nonvanishing profile pairs in ℂ² · Euclidean space ℝ⁴ · Depends on the construction

Ambient space: Euclidean space ℝ⁴
Topology: Profile-product parameter surface, with any stated period identifications
Embeddedness: Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Convex slab translators with polyhedral asymptotics

Source-informed schematic

Earliest recorded reference: 2020-06-29

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Higher-dimensional polyhedral constructions, n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Complete convex graphical hypersurface; polyhedral asymptotic data
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: One topological end.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Translators with an embedded rotational shrinking-torus end (Angenent, 1992; Sun–Zhihan Wang, 2023)

Numerical Angenent profile; auxiliary end model

Earliest recorded reference: 2023-01-19

Genus one belongs to a horizontal surface cross-section, not the three-dimensional translator. The full core topology is not classified.

Compact core with one T² × [z₀,∞) end · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: Compact core with one T² × [z₀,∞) end
Embeddedness: Embedded: The constructed three-dimensional translators in ℝ⁴ are properly embedded.
Compactness: Noncompact
Surface genus: Not a surface: translator dimension 3
Topological ends: 1 topological end
Entropy: Exact formula: λ(Σ) = λ(S_A) = F₀,₁(S_A). Exact relation to the input Angenent torus; no exact numerical value.
Gaussian area: Not recorded: Not recorded for the stated center and scale.

Translators with high-genus spherical-doubling ends (Kapouleas–McGrath, 2020; Sun–Zhihan Wang, 2023)

Interactive horizontal-slice schematic in ℝ⁴

Earliest recorded reference: 2023-01-19

The horizontal cross-section has genus 2m−1 for sufficiently large m; this is not a surface genus of the translator.

Compact core with one S_m × [z₀,∞) end · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: Compact core with one S_m × [z₀,∞) end
Embeddedness: Embedded: The constructed three-dimensional translators in ℝ⁴ are properly embedded.
Compactness: Noncompact
Surface genus: Not a surface: translator dimension 3
Topological ends: 1 topological end
Entropy: Proved bound or relation: λ(Σ_m) = λ(S_m) < 8/e ≈ 2.94304.
Gaussian area: Not recorded: Not recorded for the stated center and scale.

Translators with general closed-shrinker ends

Source-informed schematic

Earliest recorded reference: 2023-01-19

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Smooth construction, n ≤ 6, new inputs starting at n = 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Complete translator with prescribed closed-shrinker tangent flow; full topology not specified
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

n ≥ 7, possible singular set of dimension ≤ n−7 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Possibly singular minimizing translator
Embeddedness: Embedded: Embedded geometric construction; smoothness may fail on a set of dimension at most n−7.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Su translators obtained by gluing grim cylinders

Projected construction model

Earliest recorded reference: 2020-11-16

Full Euclidean ambient dimension is 2m, at least six. The illustration uses m = 3, equal angles π/3, and λ = 0; it shows selected meridians and a projected two-dimensional slice of an illustrative gluing model. Projection crossings do not contradict embeddedness in C³. The time views keep the gluing scale fixed and translate by t e₃; no final PDE correction is computed. Auxiliary slices do not determine genus or ends of this higher-dimensional example.

Two-cylinder desingularizations in Cᵐ, m ≥ 3 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Sᵐ⁻¹ × R
Embeddedness: Embedded: Embedded for this explicitly restricted case; no claim about other weights, groups, or singular limits.
Compactness: Noncompact
Surface genus: Intrinsic dimension at least three; surface genus is inapplicable.
Topological ends: Two ends from the original grim cylinders; higher-end arrangements in Corollary 1.1 are not assigned to this case.
Entropy: Not recorded: No value or bound recorded from the checked source; entropy is not inferred from a single centered Gaussian integral.
Gaussian area: Not recorded: No value or bound recorded from the checked source; entropy is not inferred from a single centered Gaussian integral.

Su–Wood Lagrangian translating helicoids

Numerical ODE profile projection

Earliest recorded reference: 2025-07-02

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

SO(m−1) case, m ≥ 3 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: G-orbits of a two-dimensional profile; quotient identifications as in source
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Su–Wood paired-profile translators

Numerical ODE profile projection

Earliest recorded reference: 2025-07-02

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

SO(m−1) case, m ≥ 3 · Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean dimension ≥ 5
Topology: G-orbits of a two-dimensional profile; quotient identifications as in source
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

T^(m−2) case, m ≥ 3 · Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean dimension ≥ 5
Topology: Torus-orbit profile construction
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Nonconvex entire graphical translators

Source-informed schematic

Earliest recorded reference: 2026-09-04

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Entire graphs, N ≥ 8, ambient ℝ^(N+1) · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: ℝᴺ, N ≥ 8
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: One topological end.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Rotational translators in the hyperbolic-plane product

Numerical ODE profile

Earliest recorded reference: 2018-03-07

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Bowl in ℍ² × ℝ · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: ℝ²
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Genus zero.
Topological ends: One topological end.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Catenoid in ℍ² × ℝ · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: S¹ × ℝ
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Genus zero.
Topological ends: Two topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Invariant translators in the Heisenberg group

Source-informed schematic

Earliest recorded reference: 2018-11-12

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Slab graphs and rotational bowl · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: ℝ²
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Genus zero.
Topological ends: One topological end.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Rotational catenoids · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: S¹ × ℝ
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Genus zero.
Topological ends: Two topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Helicoidal examples · Riemannian ambient manifold · Not embedded

Ambient space: Riemannian ambient manifold
Topology: Immersed parameter plane
Embeddedness: Not embedded: The intersecting-arm examples in the helicoidal family are nonembedded.
Compactness: Noncompact
Surface genus: Genus zero.
Topological ends: One topological end.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Invariant translators in Sol geometry

Source-informed schematic

Earliest recorded reference: 2019-06-19

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Complete graphical cases of Theorems 1.1–1.2 · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Graph over the specified slab or half-plane
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Genus zero.
Topological ends: One topological end.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Other invariant profile cases · Riemannian ambient manifold · Not recorded

Ambient space: Riemannian ambient manifold
Topology: Determined by the profile and invariant orbit
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Graphical translators over compact symmetric spaces (Fujii, 2021; Fujii–Koike, 2023)

Source-informed schematic

Earliest recorded reference: 2021-09-30

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Graphs over domains in S² · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Graph over the specified spherical domain
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Higher-dimensional compact symmetric bases · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Graph over a specified invariant open subset of N
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Hyperbolic Killing translators and catenoids (de Lima–Ramos–dos Santos, 2023; de Lima–Ramos, 2026)

Source-informed schematic

Earliest recorded reference: 2023-07-26

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Rotational annuli in ℍ³ · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: S¹ × ℝ
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Genus zero.
Topological ends: Two topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Rotational catenoids in ℍⁿ⁺¹, n ≥ 3 · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Sⁿ⁻¹ × ℝ
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Two topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Grim and helicoidal surface constructions · Riemannian ambient manifold · Not recorded

Ambient space: Riemannian ambient manifold
Topology: Depends on the specified invariant profile
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Helicoidal rotators-translators in the hyperbolic-plane product

Source-informed schematic

Earliest recorded reference: 2024-02-07

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Complete helicoidal surface in ℍ² × ℝ · Riemannian ambient manifold · Not recorded

Ambient space: Riemannian ambient manifold
Topology: Immersed parameter plane ℝ²
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Genus zero.
Topological ends: One topological end.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Grim reapers in the hyperbolic-plane product

Source-informed schematic

Earliest recorded reference: 2024-02-08

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Vertical, parabolic and hyperbolic Killing cases · Riemannian ambient manifold · Depends on the construction

Ambient space: Riemannian ambient manifold
Topology: Depends on the profile and invariant subgroup
Embeddedness: Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Parabolic Killing grim reapers in hyperbolic space

Source-informed schematic

Earliest recorded reference: 2024-02-08

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Parabolic invariant surfaces in ℍ³ · Riemannian ambient manifold · Not recorded

Ambient space: Riemannian ambient manifold
Topology: Invariant profile construction; case-dependent domain
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Invariant translators in the special linear group

Source-informed schematic

Earliest recorded reference: 2024-05-24

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Invariant surfaces in SL(2,ℝ) · Riemannian ambient manifold · Depends on the construction

Ambient space: Riemannian ambient manifold
Topology: Profile × invariant orbit, with the stated circle identifications
Embeddedness: Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness: Not specified for this subfamily
Surface genus: Not recorded.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Expanding planar networks with a bounded cell

Numerical ODE network

Earliest recorded reference: 2007

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

k ≥ 7 equal-tension networks · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: One bounded k-sided cell and k rays
Embeddedness: Embedded: Embedded network with prescribed 120-degree triple junctions.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: At least four ends, as specified in the statement.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Self-expanding planar tree networks (Mazzeo–Saez, 2007; Ilmanen–Neves–Schulze, 2014)

Numerical ODE solution

Earliest recorded reference: 2007-04-24

Networks are not smooth immersed curves at their junctions; genus is not a surface invariant here.

Three prescribed rays · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Planar tree with triple junctions
Embeddedness: Embedded: Embedded as a network: branches meet only at their prescribed triple junctions. This is not a smooth embedded curve at a junction.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 3
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Four or more prescribed rays · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Planar tree with triple junctions
Embeddedness: Embedded: Embedded as a network: branches meet only at their prescribed triple junctions. This is not a smooth embedded curve at a junction.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Number of prescribed rays (at least 4)
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Expanding smoothings of planar corners (Halldórsson, 2010; Ding, 2015)

Numerical ODE curve

Earliest recorded reference: 2010-07-09

Expanding corner curve · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Line ℝ
Embeddedness: Embedded: The graphical smoothing is properly embedded, as is its product with a line.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Product with a line · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ²
Embeddedness: Embedded: The graphical smoothing is properly embedded, as is its product with a line.
Compactness: Noncompact
Surface genus: 0
Topological ends: 1
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Graphical expanders from general Lipschitz cones

Numerical PDE approximation

Earliest recorded reference: 1989

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

n = 2 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: ℝ²
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Genus zero.
Topological ends: One topological end.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: ℝⁿ
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: One topological end.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Two-sheeted expanders of rotational double cones

Numerical ODE surface

Earliest recorded reference: 1995

Disconnected: two components. Genus is recorded componentwise.

Two disjoint planes ℝ² ⊔ ℝ² · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Two disjoint planes ℝ² ⊔ ℝ²
Embeddedness: Embedded: The two smooth graphical components are disjoint; their disconnected union is embedded.
Compactness: Noncompact
Surface genus: 0 on each component
Topological ends: 2 total; 1 on each component
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Higher-dimensional double-cone smoothings · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Two disjoint copies of ℝⁿ, n ≥ 3
Embeddedness: Embedded: The two smooth graphical components are disjoint; their disconnected union is embedded.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2 total; 1 on each component
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

One-sheeted expanders of rotational double cones

Numerical ODE surface

Earliest recorded reference: 1995

Annulus S¹ × ℝ · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Annulus S¹ × ℝ
Embeddedness: Embedded: The rotational neck is a properly embedded hypersurface.
Compactness: Noncompact
Surface genus: 0
Topological ends: 2
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Higher-dimensional necks · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sⁿ⁻¹ × ℝ, n ≥ 3
Embeddedness: Embedded: The rotational neck is a properly embedded hypersurface.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Rotational graphical expanders over round cones

Numerical ODE surface

Earliest recorded reference: 2015-03-09

Radial graph surface · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ²
Embeddedness: Embedded: The smooth entire radial graph is properly embedded.
Compactness: Noncompact
Surface genus: 0
Topological ends: 1
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Higher-dimensional radial graphs · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Euclidean space ℝⁿ, n ≥ 3
Embeddedness: Embedded: The smooth entire radial graph is properly embedded.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 1
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

One-sided expanders from mean-convex cones

Foliation schematic

Earliest recorded reference: 2015-03-09

Depends on the chosen cone and side · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Depends on the chosen cone and side
Embeddedness: Embedded: The stated smooth minimizing-expander theorem gives an embedded hypersurface on the chosen side of the cone.
Compactness: Noncompact
Surface genus: Depends on parameters
Topological ends: Depends on parameters
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Higher-dimensional cone smoothings · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Depends on the chosen cone and side
Embeddedness: Embedded: The stated smooth minimizing-expander theorem gives an embedded hypersurface on the chosen side of the cone.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Depends on parameters
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Mountain-pass self-expanders

Heuristic schematic

Earliest recorded reference: 2020-03-30

Smooth surface case n = 2 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Topology depends on the cone and variational construction
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: Not prescribed
Topological ends: Determined by the cone link; no common count
Entropy: Finite; value not recorded: Finite by smooth conical asymptotics; no value supplied.
Gaussian area: Finite; value not recorded: Finite by area growth; not equal to entropy in general.

Smooth hypersurfaces 3 ≤ n ≤ 6 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Smooth hypersurface; topology not prescribed
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Depends on the cone link
Entropy: Finite; value not recorded: Finite by smooth conical asymptotics.
Gaussian area: Finite; value not recorded: Finite by area growth.

Three-ended expanders of positive genus (Shao–Zou, 2025; Hoffman–Martín–White, 2026)

Interactive topology schematic

Earliest recorded reference: 2025-07-25

Fixed three-component cone Cₖ in ℝ³ · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Orientable genus pᵃk−1 surface, p odd prime, a ≥ 0; k sufficiently large
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: g = pᵃk−1 ≥ 2 for the stated large k
Topological ends: 3 conical ends
Entropy: Finite; value not recorded: Finite by conical area growth; no value supplied.
Gaussian area: Finite; value not recorded: Finite; no centered value supplied.

Reported multiphase expanding surface constructions

Source-informed schematic

Earliest recorded reference: 2025-12-02

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Reported multiphase construction in ℝ³ · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Not certified from the current review
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Not specified for this subfamily
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Knotted self-expanders

Topology schematic

Earliest recorded reference: 2026-11-02

Trefoil example · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Trefoil-knotted torus with three punctures and attached annular ends
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: 1
Topological ends: 3
Entropy: Finite; value not recorded: Finite by conical area growth; no value supplied.
Gaussian area: Finite; value not recorded: Finite; no centered value supplied.

General handlebody construction · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Knotted genus-g surface, every g ≥ 1, with sufficiently many ends
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: Any g ≥ 1
Topological ends: r sufficiently large for the chosen handlebody
Entropy: Finite; value not recorded: Finite by conical area growth.
Gaussian area: Finite; value not recorded: Finite by conical area growth.

Lagrangian expander necks between transverse planes (Joyce–Lee–Tsui, 2008; Imagi–Joyce–Oliveira dos Santos, 2014)

Numerical quadrature and projection

Earliest recorded reference: 2008-01-24

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Lagrangian neck in C² · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: Annulus S¹ × R
Embeddedness: Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness: Noncompact
Surface genus: 0 (annulus).
Topological ends: Two planar ends.
Entropy: Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.
Gaussian area: Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.

Lagrangian neck in Cⁿ, n ≥ 3 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Sⁿ⁻¹ × R
Embeddedness: Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness: Noncompact
Surface genus: Intrinsic dimension n ≥ 3; surface genus does not apply.
Topological ends: Two planar ends.
Entropy: Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.
Gaussian area: Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.

Noncompact periodic Lagrangian expanders

Source-informed schematic

Earliest recorded reference: 2008-01-24

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

n = 2, m = 1 · Euclidean space ℝ⁴ · Not recorded

Ambient space: Euclidean space ℝ⁴
Topology: Periodic parameter domain S¹ × S⁰ × ℝ; image identifications may occur
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Genus zero.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

n ≥ 3 · Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean dimension ≥ 5
Topology: S¹ × Sᵐ⁻¹ × ℝⁿ⁻ᵐ, as parameter domain
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Two-convex Lagrangian graphical expanders

Exact initial-cone projection

Earliest recorded reference: 2023-09-18

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Lagrangian expander graph in C² · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: R²
Embeddedness: Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness: Noncompact
Surface genus: 0 (plane topology).
Topological ends: One end of Rⁿ, n ≥ 2.
Entropy: Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Gaussian area: Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.

Lagrangian expander graph in Cⁿ, n ≥ 3 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Rⁿ
Embeddedness: Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness: Noncompact
Surface genus: Intrinsic dimension n ≥ 3; surface genus does not apply.
Topological ends: One end of Rⁿ, n ≥ 2.
Entropy: Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.
Gaussian area: Not recorded: No value or bound recorded from the checked source; centered Gaussian area and entropy are separate quantities.

Bi-rotational expanding hypersurfaces

Numerical ODE profile projection

Earliest recorded reference: 2026-10-19

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Two-ended p,q ≥ 1 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Depends on the construction

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: ℝ × Sᵖ × Sᑫ
Embeddedness: Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Two topological ends.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Axis-starting p,q ≥ 1 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: ℝ^(p+1) × Sᑫ or ℝ^(q+1) × Sᵖ
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: One topological end.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Lagrangian expanders with minimal Legendrian leaves (Anciaux, 2006; Madnick–Wood, 2022)

Numerical ODE profile and projected immersion

Earliest recorded reference: 2006

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Non-spherical compact Legendrian leaves, n ≥ 3 · Euclidean dimension ≥ 5 · Depends on the construction

Ambient space: Euclidean dimension ≥ 5
Topology: ℝ × M
Embeddedness: Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Two topological ends.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Lee–Wang embedded quadric expanders

Exact time slices and circle orbit

Earliest recorded reference: 2008-02-04

Full Euclidean ambient dimension is 2n or 2m, at least six. Auxiliary slices and curves do not determine the genus of this higher-dimensional example.

Embedded unit-weight expanders in Cⁿ, n ≥ 3, n/2 < k < n · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Mapping torus of Rᵏ × Sⁿ⁻ᵏ⁻¹ under antipodal gluing; Rᵏ × S¹ when n − k = 1.
Embeddedness: Embedded: Embedded for this explicitly restricted case; no claim about other weights, groups, or singular limits.
Compactness: Noncompact
Surface genus: Intrinsic dimension at least three; surface genus is inapplicable.
Topological ends: One end; two fiber components in the n − k = 1 case are joined by the gluing.
Entropy: Not recorded: No value or bound recorded from the checked source; entropy is not inferred from a single centered Gaussian integral.
Gaussian area: Finite; value not recorded: Finite by the explicit quadric geometry and polynomial volume growth; no value recorded.

Expanders based on generalized Lawson–Osserman maps

Numerical ODE radial profile

Earliest recorded reference: 2022-11-10

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Hopf type (3,2,2) and other admitted LOMSE types · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: ℝ^(n+1); ambient ℝ^(n+m+2), first case ℝ⁷
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: One topological end.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

The ancient sine wave (Nakayama–Iizuka–Wadati, 1994; Angenent–You, 2018)

Exact formula

Earliest recorded reference: 1994

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Complete periodic graph in ℝ² · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: ℝ with spatially periodic height
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Two topological ends.
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Purely rotating curve-shortening solitons (Halldórsson, 2010; Halldórsson, 2011; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)

Numerical ODE curve

Earliest recorded reference: 2010-07-09

All profiles are complete proper curves with two ends. A nonzero kernel projection gives an embedding; the general even-dimensional case is not assigned that conclusion. Planar properties are sharpened by Halldórsson (2012); the spatial and higher-dimensional cases retain their own hypotheses.

Complete planar rotators, including Yin–Yang · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Noncompact line ℝ
Embeddedness: Embedded: Theorem 3.1 proves proper embeddedness for every minimum radius.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2 intrinsic ends of the parameter manifold
Entropy: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.
Gaussian area: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.

Nonplanar rotators in R3 with nonzero axial tangent · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Line R (immersed domain)
Embeddedness: Embedded: The axial coordinate is strictly monotone (Lemma 5.4), and both ends escape (Lemma 6.3): a proper embedding.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Rotators in R4 with invertible rotation map · Euclidean space ℝ⁴ · Not recorded

Ambient space: Euclidean space ℝ⁴
Topology: Line R (immersed domain)
Embeddedness: Not recorded: No general embeddedness theorem for the stated initial data is recorded in the reviewed source.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Rotators in R^n, n ≥ 5, with nonconstant kernel projection · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Line R (immersed domain)
Embeddedness: Embedded: Projection onto a line in ker A is injective; properness follows from escaping ends.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Rotators in R^(2m), m ≥ 3, with invertible rotation map · Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean dimension ≥ 5
Topology: Line R (immersed domain)
Embeddedness: Not recorded: No general embeddedness theorem for the stated initial data is recorded in the reviewed source.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Rotating product surfaces · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ² (curve × ℝ)
Embeddedness: Embedded: The product of a properly embedded rotating planar curve with ℝ is properly embedded.
Compactness: Noncompact
Surface genus: 0
Topological ends: One end of the parameter plane
Entropy: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.
Gaussian area: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.

Rotating and expanding curve solitons (Halldórsson, 2010; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)

Numerical ODE curve

Earliest recorded reference: 2010-07-09

α > 0; complete proper immersed curves. Generalized logarithmic spirals are end asymptotes, not exact spatial formulas. No pure-expander Gaussian identity is imposed. Planar properties are sharpened by Halldórsson (2012); the spatial and higher-dimensional cases retain their own hypotheses.

A ≠ 0, B > 0 · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Noncompact line ℝ
Embeddedness: Embedded: Properly embedded for every minimum radius by Theorem 3.1.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2 intrinsic ends of the parameter manifold
Entropy: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.
Gaussian area: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.

Spatial rotating expanders · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Line R (immersed domain)
Embeddedness: Not recorded: No general embeddedness theorem for the stated initial data is recorded in the reviewed source.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Rotating expanders in R4 · Euclidean space ℝ⁴ · Not recorded

Ambient space: Euclidean space ℝ⁴
Topology: Line R (immersed domain)
Embeddedness: Not recorded: No general embeddedness theorem for the stated initial data is recorded in the reviewed source.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Rotating expanders in R^n, n ≥ 5 · Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean dimension ≥ 5
Topology: Line R (immersed domain)
Embeddedness: Not recorded: No general embeddedness theorem for the stated initial data is recorded in the reviewed source.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Rotating and shrinking curve solitons (Halldórsson, 2010; D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012)

Numerical ODE curve

Earliest recorded reference: 2010-07-09

α < 0, P_T(AC) not identically zero. All recorded domains are complete, noncompact and nonproper; bounded image does not mean compact domain. The spherical subfamily is Euclidean flow, not a separate curved-ambient entry. End convergence to a single circle requires a two-dimensional frequency eigenspace. Unresolved kernel-limit scenarios are not existence claims. Planar properties are sharpened by Halldórsson (2012); the spatial and higher-dimensional cases retain their own hypotheses.

Bounded two-circular-end profiles · Euclidean plane ℝ² · Not recorded

Ambient space: Euclidean plane ℝ²
Topology: Noncompact line ℝ
Embeddedness: Not recorded: The source records self-intersecting members but no complete embeddedness classification; tentative figure captions do not prove embeddedness.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2 intrinsic ends of the parameter manifold
Entropy: Exact formula: ∞ at every regular time, with arclength counted on the immersion: infinite length in a compact region makes every Gaussian integral diverge.
Gaussian area: Exact formula: ∞ at every regular time, with arclength counted on the immersion: infinite length in a compact region makes every Gaussian integral diverge.

Comet spirals · Euclidean plane ℝ² · Not recorded

Ambient space: Euclidean plane ℝ²
Topology: Noncompact line ℝ
Embeddedness: Not recorded: Existence is proved; uniqueness and a complete embeddedness criterion are not supplied. Some figures show crossings and others ask whether the curve is embedded.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2 intrinsic ends of the parameter manifold
Entropy: Exact formula: ∞ at every regular time, with arclength counted on the immersion: infinite length in a compact region makes every Gaussian integral diverge.
Gaussian area: Exact formula: ∞ at every regular time, with arclength counted on the immersion: infinite length in a compact region makes every Gaussian integral diverge.

Genuinely rotating shrinking curves in R3 · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Line R (immersed domain)
Embeddedness: Not recorded: No general embeddedness theorem for the stated initial data is recorded in the reviewed source.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two intrinsic ends; at least one accumulates in a bounded region
Entropy: Exact formula: λ = +∞: already every positive-scale Gaussian integral diverges on a bounded infinite-length tail.
Gaussian area: Exact formula: F₀,₁ = +∞: infinite arclength in a bounded region (direct calculation).

Tangent-sphere rotating curves in R3 · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Line R (immersed domain)
Embeddedness: Not recorded: No general embeddedness theorem for the stated initial data is recorded in the reviewed source.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two intrinsic ends asymptotic to the equator, with opposite orientations
Entropy: Exact formula: λ = +∞: already every positive-scale Gaussian integral diverges on a bounded infinite-length tail.
Gaussian area: Exact formula: F₀,₁ = +∞: infinite arclength in a bounded region (direct calculation).

Genuinely rotating shrinking curves in R4 · Euclidean space ℝ⁴ · Not recorded

Ambient space: Euclidean space ℝ⁴
Topology: Line R (immersed domain)
Embeddedness: Not recorded: No general embeddedness theorem for the stated initial data is recorded in the reviewed source.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Exact formula: λ = +∞: already every positive-scale Gaussian integral diverges on a bounded infinite-length tail.
Gaussian area: Exact formula: F₀,₁ = +∞: infinite arclength in a bounded region (direct calculation).

Genuinely rotating shrinking curves in R^n, n ≥ 5 · Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean dimension ≥ 5
Topology: Line R (immersed domain)
Embeddedness: Not recorded: No general embeddedness theorem for the stated initial data is recorded in the reviewed source.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Exact formula: λ = +∞: already every positive-scale Gaussian integral diverges on a bounded infinite-length tail.
Gaussian area: Exact formula: F₀,₁ = +∞: infinite arclength in a bounded region (direct calculation).

The Angenent oval as an explicit ancient flow

Exact implicit solution

Earliest recorded reference: 2011-06-01

The entry does not currently give a slice-by-slice entropy value.

Circle S¹ · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Circle S¹
Embeddedness: Embedded: Every time slice for t < 0 is an embedded strictly convex closed curve.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite; no value recorded.
Gaussian area: Not recorded: Not recorded in this entry.

Eternal curves from chains of grim reapers (Angenent–You, 2018; Choi–Seo–Su–Zhao, 2026)

Numerical curve-shortening flow

Earliest recorded reference: 2018-03-04

Monotone three-reaper chain · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Line ℝ
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2
Entropy: Proved bound or relation: Each slice has entropy at most 4; the supremum over the whole selected m = 3 eternal flow is exactly 4 by Choi–Seo–Su–Zhao (2026), Corollary 1.5. Equality for every slice is not asserted.
Gaussian area: Not recorded: No fixed-center Gaussian-area value recorded; it is not identified with entropy.

Immersed ancient curves from grim-reaper chains

Source-informed schematic

Earliest recorded reference: 2018-03-04

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Compact immersed chain · Euclidean plane ℝ² · Depends on the construction

Ambient space: Euclidean plane ℝ²
Topology: S¹
Embeddedness: Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Noncompact immersed chain · Euclidean plane ℝ² · Depends on the construction

Ambient space: Euclidean plane ℝ²
Topology: ℝ with two asymptotic lines
Embeddedness: Depends on the construction: The status depends on the chosen input and closing or quotient conditions.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Two topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Immersed ancient curves spanning a half-plane

Source-informed schematic

Earliest recorded reference: 2020-03-06

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Compact immersed ancient curves · Euclidean plane ℝ² · Depends on the construction

Ambient space: Euclidean plane ℝ²
Topology: S¹ as parameter manifold
Embeddedness: Depends on the construction: Early slices may self-intersect; the constructed flow becomes embedded later.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Convex ancient free-boundary flows in balls (Bourni–Langford, 2021; Bourni–Langford, 2022)

Source-informed schematic

Earliest recorded reference: 2021-11-11

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Curve in B² · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Closed interval with endpoints on ∂B²
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Disk in B³ · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: D² with boundary on ∂B³
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Genus zero.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Dⁿ in Bⁿ⁺¹
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Compact nonconvex ancient spiral curves

Numerical ODE with asymptotic cap

Earliest recorded reference: 2022-04-11

The time-dependent figure shows numerical rotating-soliton arms with an asymptotic cap. Its displayed times and geometry are model values, not a numerical construction or certification of the ancient solution.

Embedded ancient planar curve · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Circle S¹
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite for each smooth compact slice; no uniform value over ancient time is asserted.
Gaussian area: Finite; value not recorded: Finite for each smooth compact slice; no uniform value over ancient time is asserted.

Ancient curves with mixed boundary conditions

Source-informed schematic

Earliest recorded reference: 2024-04-01

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

0 < d < 1, eternal case · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Interval with one fixed and one free endpoint
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

d = 1, finite-time endpoint singularity · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Interval before the singular time
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Ancient free-boundary curves in convex domains

Source-informed schematic

Earliest recorded reference: 2024-04-12

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Planar strictly convex domain · Euclidean plane ℝ² · Embedded

Ambient space: Euclidean plane ℝ²
Topology: Compact interval with free endpoints on the fixed boundary
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Minimal surfaces as stationary mean curvature flows (Chopp, 1993; Mantegazza, 2011)

Exact formula

Earliest recorded reference: 1993

Every smooth minimal immersion in a fixed ambient metric gives a stationary flow for all real times. The cases below concern immersions without boundary. Completeness, properness, symmetry and stability depend on the chosen immersion; none is assumed for the general class. The catenoid is the illustrated example. Genus describes the parameter surface when orientable; the fixed-boundary numerical catenoid benchmark is discussed in the text.

Catenoid in ℝ³ (illustrated example) · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Annulus S¹ × ℝ
Embeddedness: Embedded: Every stationary time slice is a properly embedded annulus.
Compactness: Noncompact
Surface genus: 0
Topological ends: 2
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded in this entry.

Affine planes in ℝ³ · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: ℝ²; complete and properly embedded
Embeddedness: Embedded: The stated affine parametrization is an embedding; multiple covers are not included in this case.
Compactness: Noncompact
Surface genus: 0
Topological ends: 1
Entropy: Exact formula: λ(P) = 1 for every multiplicity-one affine plane.
Gaussian area: Exact formula: F₀,₁(P) = exp(−dist(0,P)²/4); equals 1 exactly when the plane contains the origin.

General minimal surfaces in ℝᴺ, N ≥ 3 (codimension N − 2) · Euclidean space ℝ³; Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Depends on the construction

Ambient space: Euclidean space ℝ³; Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: The topology of the chosen minimal parameter manifold; completeness and properness are not assumed.
Embeddedness: Depends on the construction: The stationary flow is embedded exactly when the chosen minimal immersion is an embedding. Nonembedded immersions remain nonembedded; minimality alone does not decide this.
Compactness: Noncompact
Surface genus: Depends on the chosen orientable parameter surface; no single genus is assigned to this class.
Topological ends: Depends on the chosen minimal immersion; no common number is asserted.
Entropy: Not recorded: No value, uniform bound or general finiteness assertion is recorded.
Gaussian area: Not recorded: No value or general finiteness assertion is recorded; not identified with entropy.

Minimal submanifolds of dimension n ≥ 3 in ℝᴺ, N ≥ n + 1 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Depends on the construction

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: The topology of the chosen minimal parameter manifold; completeness and properness are not assumed.
Embeddedness: Depends on the construction: The stationary flow is embedded exactly when the chosen minimal immersion is an embedding. Nonembedded immersions remain nonembedded; minimality alone does not decide this.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Depends on the chosen minimal immersion; no common number is asserted.
Entropy: Not recorded: No value, uniform bound or general finiteness assertion is recorded.
Gaussian area: Not recorded: No value or general finiteness assertion is recorded; not identified with entropy.

Minimal surfaces in fixed Riemannian manifolds of dimension N ≥ 3 · Riemannian ambient manifold · Depends on the construction

Ambient space: Riemannian ambient manifold
Topology: The topology of the chosen minimal parameter manifold; completeness and properness are not assumed.
Embeddedness: Depends on the construction: The stationary flow is embedded exactly when the chosen minimal immersion is an embedding. Nonembedded immersions remain nonembedded; minimality alone does not decide this.
Compactness: Not specified for this subfamily
Surface genus: Depends on the chosen orientable parameter surface; no single genus is assigned to this class.
Topological ends: Depends on the chosen minimal immersion; no common number is asserted.
Entropy: Not applicable: Euclidean entropy is not assigned to intrinsic MCF in a curved ambient manifold.
Gaussian area: Not applicable: Euclidean Gaussian area is not assigned to intrinsic MCF in a curved ambient manifold.

Minimal submanifolds of dimension n ≥ 3 in fixed Riemannian manifolds of dimension N ≥ n + 1 · Riemannian ambient manifold · Depends on the construction

Ambient space: Riemannian ambient manifold
Topology: The topology of the chosen minimal parameter manifold; completeness and properness are not assumed.
Embeddedness: Depends on the construction: The stationary flow is embedded exactly when the chosen minimal immersion is an embedding. Nonembedded immersions remain nonembedded; minimality alone does not decide this.
Compactness: Not specified for this subfamily
Surface genus: Not a surface
Topological ends: Depends on the chosen minimal immersion; no common number is asserted.
Entropy: Not applicable: Euclidean entropy is not assigned to intrinsic MCF in a curved ambient manifold.
Gaussian area: Not applicable: Euclidean Gaussian area is not assigned to intrinsic MCF in a curved ambient manifold.

Rotational degenerate neckpinch solutions

Asymptotic time sequence with numerical bowl cap

Earliest recorded reference: 1997

Topology is for regular time slices before the singular time.

Surface neckpinches · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Sphere S²
Embeddedness: Embedded: Every regular slice before the singular time is an embedded sphere; the singular limiting profile is not the classified slice.
Compactness: Compact
Surface genus: 0
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite; no value recorded.
Gaussian area: Not recorded: Not recorded in this entry.

Higher-dimensional neckpinches · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sphere Sⁿ, n ≥ 3
Embeddedness: Embedded: Every regular slice before the singular time is an embedded sphere; the singular limiting profile is not the classified slice.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite; no value recorded.
Gaussian area: Not recorded: Not recorded in this entry.

Helicoidal minimal surfaces as stationary flows

Exact formula

Earliest recorded reference: 2011-06-22

Full parameter plane, not its quotient by vertical periodicity. The nonzero-CMC comparison family is not asserted to be an MCF soliton.

Classical helicoid (a = 0, h > 0) · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Plane ℝ²
Embeddedness: Embedded: The exact helicoid parametrization is a proper embedding.
Compactness: Noncompact
Surface genus: 0
Topological ends: One end of the parameter plane
Entropy: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.
Gaussian area: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.

Nonzero-offset helicoidal minimal immersions (a ≠ 0, h > 0) · Euclidean space ℝ³ · Not embedded

Ambient space: Euclidean space ℝ³
Topology: Parameter plane ℝ²
Embeddedness: Not embedded: The generating curve has infinitely many actual self-intersections; these produce intersections at the same height on the surface.
Compactness: Noncompact
Surface genus: 0
Topological ends: One end of the parameter plane
Entropy: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.
Gaussian area: Not recorded: No value or bound recorded in the reviewed source passages; centered Gaussian area is not identified with entropy.

Eternal shrinking helices (D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012; Stryker–Sun, 2019)

Exact R3 curve

Earliest recorded reference: 2012-07-17

Entropy depends on time; infinity is its supremum over the eternal time interval, not a value at any finite time.

R3 helix · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Line ℝ
Embeddedness: Embedded: The last coordinate equals the real parameter, making every finite-time slice a proper embedding.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2 winding ends
Entropy: Exact formula: λ(Γₜ) = √(1+D(t)), D = Σⱼ kⱼ² r^(2kⱼ²). Finite at each time; supₜ λ = ∞. Direct Gaussian integration.
Gaussian area: Exact formula: F₀,₁(Γₜ) = √(1+D(t)) e^(−B(t)/4), B = Σⱼ r^(2kⱼ²).

Higher-codimension helices · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Line ℝ in ℝ²ᵐ⁺¹, m ≥ 2
Embeddedness: Embedded: The last coordinate equals the real parameter, making every finite-time slice a proper embedding.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2 winding ends
Entropy: Exact formula: λ(Γₜ) = √(1+D(t)), D = Σⱼ kⱼ² r^(2kⱼ²). Finite at each time; supₜ λ = ∞. Direct Gaussian integration.
Gaussian area: Exact formula: F₀,₁(Γₜ) = √(1+D(t)) e^(−B(t)/4), B = Σⱼ r^(2kⱼ²).

Rotating and translating curve solitons

Numerical ODE curve

Earliest recorded reference: 2012-07-17

A ≠ 0, v ≠ 0, Av = 0, with genuinely nonzero normal rotation. The rotation map must have a kernel. Curves lie in range(A) plus at most a two-dimensional affine kernel component (Lemma 5.2); R4 permits a two-dimensional kernel component.

Monotone-height screw solitons in R3 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Line R (immersed domain)
Embeddedness: Embedded: Bijective height projection makes the complete proper curve an embedding.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Screw solitons with a height minimum in R3 · Euclidean space ℝ³ · Not recorded

Ambient space: Euclidean space ℝ³
Topology: Line R (immersed domain)
Embeddedness: Not recorded: Each branch is a height graph, but disjointness of the two branches is not established by that fact.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Monotone-height screw solitons in R4 · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: Line R (immersed domain)
Embeddedness: Embedded: Bijective height projection makes the complete proper curve an embedding.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Screw solitons with a height minimum in R4 · Euclidean space ℝ⁴ · Not recorded

Ambient space: Euclidean space ℝ⁴
Topology: Line R (immersed domain)
Embeddedness: Not recorded: Each branch is a height graph, but disjointness of the two branches is not established by that fact.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Monotone-height screw solitons in R^n, n ≥ 5 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Line R (immersed domain)
Embeddedness: Embedded: Bijective height projection makes the complete proper curve an embedding.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Screw solitons with a height minimum in R^n, n ≥ 5 · Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean dimension ≥ 5
Topology: Line R (immersed domain)
Embeddedness: Not recorded: Each branch is a height graph, but disjointness of the two branches is not established by that fact.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two ends of the parameter manifold R
Entropy: Not recorded: No entropy value or finiteness theorem is recorded for this general case.
Gaussian area: Not recorded: No centered Gaussian length value is recorded; it is not identified with entropy.

Compact noncollapsed convex ancient ovals (Haslhofer–Hershkovits, 2013; Huisken–Sinestrari, 2014)

Numerical PDE approximation

Earliest recorded reference: 2013-08-19

The time-dependent illustration computes finite-seed rotational MCF for n = 2, j = 1. It is not certified ancient-flow data and does not supply Gaussian-area or entropy values. The properties below refer to the mathematical ancient flows.

Surface oval n = 2, j = 1 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Sphere S²
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Compact
Surface genus: 0
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite for each smooth compact slice; no uniform value over ancient time is asserted.
Gaussian area: Finite; value not recorded: Finite for each smooth compact slice; no uniform value over ancient time is asserted.

Hypersurface ovals n ≥ 3, 1 ≤ j ≤ n−1 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sphere Sⁿ
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite for each smooth compact slice; no uniform value over ancient time is asserted.
Gaussian area: Finite; value not recorded: Finite for each smooth compact slice; no uniform value over ancient time is asserted.

Rotational collapsed ancient pancakes

Source-informed schematic

Earliest recorded reference: 2017-05-19

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Surface pancake, n = 2 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: S²
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Genus zero.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sⁿ
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Ancient flows from unstable minimal hypersurfaces (Melko, 2010; Mramor–Payne, 2019; Choi–Huang–Lee, 2024)

Numerical Costa surface and linearized time model

Earliest recorded reference: 2019-04-17

The interactive time sequence is a numerical linearized model on a truncated Costa domain. Its Dirichlet eigenvalue and magnified normal offsets are illustration data, not additional properties of the nonlinear ancient flows.

Early ancient Costa perturbations in ℝ³ · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Genus-one surface with three ends (normal-graph topology)
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: 1
Topological ends: 3
Entropy: Not recorded: No entropy value recorded in the checked statement.
Gaussian area: Not recorded: No fixed-center Gaussian-area value recorded; it is not identified with entropy.

The eternal reapernoid

Numerical rotational mean curvature flow

Earliest recorded reference: 2019-04-17

Numerical time views approximate the paper’s outward-offset construction on truncated domains in n=2 and n=3. Surface rotation is shown only for n=2 in ℝ³. The n=3 case in ℝ⁴ is a meridian. Forward limits are recentered pointed limits: a line for n=2, a grim reaper for n≥3.

Surface reapernoid n = 2 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Annulus S¹ × ℝ
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: 0
Topological ends: 2
Entropy: Not recorded: No entropy value recorded in the checked statement.
Gaussian area: Not recorded: No fixed-center Gaussian-area value recorded; it is not identified with entropy.

Hypersurface reapernoids n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sⁿ⁻¹ × ℝ
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: 2
Entropy: Not recorded: No entropy value recorded in the checked statement.
Gaussian area: Not recorded: No fixed-center Gaussian-area value recorded; it is not identified with entropy.

Non-self-similar ancient doughnuts

Source-informed schematic

Earliest recorded reference: 2019-11-13

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Ancient torus surface in ℝ³ · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: S¹ × S¹
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Genus one.
Topological ends: No ends (compact parameter manifold).
Entropy: Proved bound or relation: λ(M_t) < 2 for the construction based on the Drugan–Nguyen shrinker; no exact value.
Gaussian area: Proved bound or relation: F₀,₁(M_t) ≤ λ(M_t) < 2 for the specified construction; not identified with entropy.

n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: S¹ × Sⁿ⁻¹
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Proved bound or relation: λ(M_t) < 2 for the construction based on the Drugan–Nguyen shrinker; no exact value.
Gaussian area: Proved bound or relation: F₀,₁(M_t) ≤ λ(M_t) < 2 for the specified construction; not identified with entropy.

Ancient pancakes with polyhedral asymptotics

Source-informed schematic

Earliest recorded reference: 2020-06-29

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

n = 2, polygonal squash-down · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: S²
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Genus zero.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sⁿ
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Convex eternal flows that do not translate

Source-informed schematic

Earliest recorded reference: 2020-06-29

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Hypersurface case n = 2 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Noncompact convex surface; global topology not recorded here
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Noncompact convex hypersurface
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Nonplanar eternal curves from grim reapers

Source-informed schematic

Earliest recorded reference: 2022-01-06

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Nonplanar curve in ℝ³ · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: ℝ
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Two topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Non-self-similar flows emerging from cones

Source-informed schematic

Earliest recorded reference: 2022-12-21

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Smooth embedded AC surface input, n = 2 · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Topology inherited from the chosen expander
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Depends on the construction.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Higher-dimensional smooth inputs · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Topology inherited from the chosen expander
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Depends on the input; asymptotic wings are not automatically topological ends.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Immortal annuli converging to a double plane

Schematic time sequence

Earliest recorded reference: 2023-12-29

Smooth for positive time; the initial trace in the construction is locally Lipschitz. The double-plane limit occurs locally as t→∞.

Annulus S¹ × ℝ · Euclidean space ℝ³ · Embedded

Ambient space: Euclidean space ℝ³
Topology: Annulus S¹ × ℝ
Embeddedness: Embedded: Every smooth positive-time slice is embedded. The multiplicity-two plane is an infinite-time limit, not a regular slice.
Compactness: Noncompact
Surface genus: 0
Topological ends: 2 graphical ends
Entropy: Not recorded: Not recorded in this entry.
Gaussian area: Not recorded: Not recorded for the stated center and scale.

Stationary special Lagrangian necks (Joyce–Lee–Tsui, 2008; Imagi–Joyce–Oliveira dos Santos, 2014)

Exact four-dimensional projection

Earliest recorded reference: 2008-01-24

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Lagrangian neck in C² · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: Annulus S¹ × R
Embeddedness: Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness: Noncompact
Surface genus: 0 (annulus).
Topological ends: Two planar ends.
Entropy: Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.
Gaussian area: Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.

Lagrangian neck in Cⁿ, n ≥ 3 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Sⁿ⁻¹ × R
Embeddedness: Embedded: The complete image is embedded in Cⁿ; crossings in a projection do not count as self-intersections.
Compactness: Noncompact
Surface genus: Intrinsic dimension n ≥ 3; surface genus does not apply.
Topological ends: Two planar ends.
Entropy: Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.
Gaussian area: Finite; value not recorded: Finite by the stated compactness or planar-end geometry; no value recorded. Centered Gaussian area is not identified with entropy unless the normalized-shrinker hypotheses apply.

Ancient torus curves in higher codimension (D. J. Altschuler–S. J. Altschuler–Angenent–Wu, 2012; Stryker–Sun, 2019)

Exact R4 curve, projected to R3

Earliest recorded reference: 2012-07-17

The chosen subfamily has k₁=1 and is embedded. A time supremum is not the exact entropy of every slice.

R4 frequencies (1,2) · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: Embedded circle S¹
Embeddedness: Embedded: The recorded subfamily has k₁ = 1, so its first two coordinates determine the circle parameter and every finite-time slice is embedded. Higher-multiplicity limits do not change this.
Compactness: Compact
Surface genus: Not a surface (a curve on a torus)
Topological ends: 0 (compact)
Entropy: Proved bound or relation: For frequencies (1,2), λ(γₜ) ≤ 2√(2π/e) ≈ 3.04069; equality is the supremum over ancient time.
Gaussian area: Exact formula: F₀,₁(γₜ) = √π √D(t) e^(t/2), D = Σⱼ kⱼ² r^(2kⱼ²). Depends on the slice t < 0.

R4 frequencies (1,k), k ≥ 2 · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: Embedded circle S¹
Embeddedness: Embedded: The recorded subfamily has k₁ = 1, so its first two coordinates determine the circle parameter and every finite-time slice is embedded. Higher-multiplicity limits do not change this.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Proved bound or relation: sup over t < 0 of λ(γₜ) = k_max √(2π/e); each slice is bounded by this. The value at a finite time is not recorded.
Gaussian area: Exact formula: F₀,₁(γₜ) = √π √D(t) e^(t/2), D = Σⱼ kⱼ² r^(2kⱼ²). Depends on the slice t < 0.

Higher-codimension torus curves · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Embedded circle S¹ in ℝ²ᵐ, m ≥ 3
Embeddedness: Embedded: The recorded subfamily has k₁ = 1, so its first two coordinates determine the circle parameter and every finite-time slice is embedded. Higher-multiplicity limits do not change this.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Proved bound or relation: sup over t < 0 of λ(γₜ) = k_max √(2π/e); each slice is bounded by this. The value at a finite time is not recorded.
Gaussian area: Exact formula: F₀,₁(γₜ) = √π √D(t) e^(t/2), D = Σⱼ kⱼ² r^(2kⱼ²). Depends on the slice t < 0.

Ancient dense torus curves with irrational frequencies

Exact R4 curve, projected to R3

Earliest recorded reference: 2012-07-17

Only even ambient dimensions 2m ≥ 4 are recorded. The closure is an m-torus, while the flow has intrinsic dimension one. Its infinite Gaussian length counts the immersed domain.

Two rationally independent frequencies in R4 · Euclidean space ℝ⁴ · Not embedded

Ambient space: Euclidean space ℝ⁴
Topology: Line R (immersed domain)
Embeddedness: Not embedded: Injective but recurrent: the inverse onto its image is not continuous. This is a nonproper immersion, not an embedding.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two intrinsic ends; both remain in a bounded torus
Entropy: Exact formula: λ = +∞: already every positive-scale Gaussian integral diverges on a bounded infinite-length tail.
Gaussian area: Exact formula: F₀,₁ = +∞: infinite arclength in a bounded region (direct calculation).

m rationally independent frequencies in R^(2m), m ≥ 3 · Euclidean dimension ≥ 5 · Not embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Line R (immersed domain)
Embeddedness: Not embedded: Injective dense torus orbit with noncontinuous inverse and nonproper image parametrization.
Compactness: Noncompact
Surface genus: Not a surface
Topological ends: Two intrinsic ends; neither escapes to infinity
Entropy: Exact formula: λ = +∞: already every positive-scale Gaussian integral diverges on a bounded infinite-length tail.
Gaussian area: Exact formula: F₀,₁ = +∞: infinite arclength in a bounded region (direct calculation).

Mean curvature evolution of the Whitney sphere

Numerical PDE evolution from exact initial data

Earliest recorded reference: 2018-02-06

Topology refers to the compact domain; the initial immersion has a transverse double point.

Whitney two-sphere · Euclidean space ℝ⁴ · Not embedded

Ambient space: Euclidean space ℝ⁴
Topology: Immersed sphere S²
Embeddedness: Not embedded: The initial Whitney immersion has a transverse double point at the origin. The regular flow remains a Lagrangian sphere and cannot become embedded.
Compactness: Compact
Surface genus: 0
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite; no value recorded.
Gaussian area: Not recorded: Not recorded in this entry.

Higher-dimensional Whitney spheres · Euclidean dimension ≥ 5 · Not embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Immersed sphere Sⁿ in ℝ²ⁿ, n ≥ 3
Embeddedness: Not embedded: The initial Whitney immersion has a transverse double point at the origin. The regular flow remains a Lagrangian sphere and cannot become embedded.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite; no value recorded.
Gaussian area: Not recorded: Not recorded in this entry.

Ancient isoparametric flows in Euclidean space

Exact formula in a labeled projection

Earliest recorded reference: 2019-11-28

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Exact product-torus case · Euclidean space ℝ⁴ · Embedded

Ambient space: Euclidean space ℝ⁴
Topology: S¹ × S¹ in ℝ⁴
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Genus one.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Other compact isoparametric leaves · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Regular isoparametric leaf; dimension and ambient codimension depend on input
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Ancient ovals with reduced symmetry

Source-informed schematic

Earliest recorded reference: 2021-05-28

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

2 ≤ k ≤ n−1, n ≥ 3 · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sⁿ
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.
Gaussian area: Finite; value not recorded: Finite on each stated smooth slice; no numerical value or uniform time bound is recorded.

Ancient stacked pancakes

Schematic 3D section

Earliest recorded reference: 2025-09-30

n = 3 in ℝ⁴, and n ≥ 4 in ℝⁿ⁺¹ · Euclidean space ℝ⁴; Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean space ℝ⁴; Euclidean dimension ≥ 5
Topology: Sphere Sⁿ, n ≥ 3, from the closed rotational profile
Embeddedness: Embedded: Embedded regular slices in the stated ambient space; schematic projections do not determine embeddedness.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Finite; value not recorded: Finite for each smooth compact slice; no uniform value over ancient time is asserted.
Gaussian area: Finite; value not recorded: Finite for each smooth compact slice; no uniform value over ancient time is asserted.

Hypersurface singularities with bounded mean curvature

Source-informed schematic

Earliest recorded reference: 2020-03-13

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Ambient ℝᴺ, N ≥ 8 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Noncompact hypersurface; minimal-cone desingularization geometry
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Lagrangian singularities with prescribed orbit models

Source-informed schematic

Earliest recorded reference: 2022-08-02

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Admissible non-spherical orbits, n ≥ 3 · Euclidean dimension ≥ 5 · Not recorded

Ambient space: Euclidean dimension ≥ 5
Topology: Cohomogeneity-one G-orbit flow, with constant isotropy type
Embeddedness: Not recorded: The reviewed statement gives an immersion; no general embedding assertion is recorded.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Not recorded for the full parameter manifold.
Entropy: Not recorded: No quantitative conclusion is recorded from the cited construction.
Gaussian area: Not recorded: No quantitative conclusion is recorded from the cited construction.

Su–Wood stationary products of Lawlor profiles

Exact formula; projected profile and circle orbits

Earliest recorded reference: 2025-07-02

Full Euclidean ambient dimension is 2m, at least six. The interactive figure uses the exact m = 3 formula, a projected two-dimensional profile, and sampled SO(2) orbits in R6. Every flow time has identical geometry. Neither the profile nor its projection determines a surface genus for this higher-dimensional example.

SO(m − 1)-invariant stationary examples in Cᵐ, m ≥ 3 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Sᵐ⁻² × R²
Embeddedness: Embedded: Embedded for this explicitly restricted case; no claim about other weights, groups, or singular limits.
Compactness: Noncompact
Surface genus: Intrinsic dimension at least three; surface genus is inapplicable.
Topological ends: One end; this is not a product of two complete Lawlor neck manifolds.
Entropy: Not recorded: No value or bound recorded from the checked source; entropy is not inferred from a single centered Gaussian integral.
Gaussian area: Not recorded: No value or bound recorded from the checked source; entropy is not inferred from a single centered Gaussian integral.

Stolarski–Su finite-time Lagrangian neck pinches

Asymptotic time illustration

Earliest recorded reference: 2026-07-03

Full Euclidean ambient dimension is 2n or 2m, at least six. Auxiliary slices and curves do not determine the genus of this higher-dimensional example.

SO(n)-invariant Type II flows: n = 3, K = 2,3,4; n ≥ 4, integer K ≥ 2 · Euclidean dimension ≥ 5 · Embedded

Ambient space: Euclidean dimension ≥ 5
Topology: Sⁿ⁻¹ × R at every regular time
Embeddedness: Embedded: Embedded for this explicitly restricted case; no claim about other weights, groups, or singular limits.
Compactness: Noncompact
Surface genus: Intrinsic dimension at least three; surface genus is inapplicable.
Topological ends: Two ends; the singular pair of planes is only a limit.
Entropy: Not recorded: No value or bound recorded from the checked source; entropy is not inferred from a single centered Gaussian integral.
Gaussian area: Not recorded: No value or bound recorded from the checked source; entropy is not inferred from a single centered Gaussian integral.

Ancient isoparametric flows in spheres

Exact formula in a labeled projection

Earliest recorded reference: 2019-11-28

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Exact Clifford-torus case · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: S¹ × S¹ in S³
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Genus one.
Topological ends: No ends (compact parameter manifold).
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Other compact isoparametric leaves · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Regular isoparametric leaf; dimension and ambient codimension depend on input
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Eternal tori in perturbations of the round three-sphere

Source-informed schematic

Earliest recorded reference: 2020-03-31

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Embedded tori in (S³,e^{2δu}g_round) · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: S¹ × S¹
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Genus one.
Topological ends: No ends (compact parameter manifold).
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Isometric curve-shortening solitons in the hyperbolic plane

Source-informed schematic

Earliest recorded reference: 2021-01-19

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Complete isometric solitons in ℍ² · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: ℝ with two ends
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: Two topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Finite-time Lagrangian neck pinches with circle symmetry

Source-informed schematic

Earliest recorded reference: 2023-05-09

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Compact circle-invariant surfaces before a pinch · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Circle fibration over an arc; smooth connected compact case
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Genus zero.
Topological ends: No ends (compact parameter manifold).
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Noncompact two-plane-asymptotic cases · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Circle-invariant surface with prescribed planar ends
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Noncompact
Surface genus: Not recorded.
Topological ends: Two topological ends.
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Lagrangian torus necks collapsing at infinite time

Exact local profiles; asymptotic time model

Earliest recorded reference: 2024-01-04

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Lagrangian connected sum in Cᵐ/Γ, m ≥ 3 · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Tᵐ # Tᵐ for every finite-time slice
Embeddedness: Embedded: Finite-time slices are embedded; the transverse double point belongs only to the infinite-time limiting union.
Compactness: Compact
Surface genus: Dimension m ≥ 3; not a surface genus.
Topological ends: No ends (compact).
Entropy: Not applicable: Intrinsic mean curvature flow in a Riemannian ambient manifold; Euclidean Gaussian area and entropy are not assigned.
Gaussian area: Not applicable: Intrinsic mean curvature flow in a Riemannian ambient manifold; Euclidean Gaussian area and entropy are not assigned.

Immortal compact flows converging to a double sphere

Qualitative time sequence

Earliest recorded reference: 2024-02-07

The flow stays in a narrow product band and also fits in a closed Sⁿ × S¹ product. Its Euclidean embedding is not an unforced Euclidean MCF.

Surface flow in S² × ℝ · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Torus S¹ × S¹ in S² × ℝ
Embeddedness: Embedded: Each smooth finite-time slice is embedded in the product ambient manifold. The multiplicity-two sphere is only the infinite-time limit.
Compactness: Compact
Surface genus: 1
Topological ends: 0 (compact)
Entropy: Not applicable: Intrinsic MCF in a curved product manifold; Euclidean entropy / Gaussian area is not assigned.
Gaussian area: Not applicable: Intrinsic MCF in a curved product manifold; Euclidean entropy / Gaussian area is not assigned.

Higher-dimensional product-manifold flows · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: S¹ × Sⁿ⁻¹ in Sⁿ × ℝ, n ≥ 3
Embeddedness: Embedded: Each smooth finite-time slice is embedded in the product ambient manifold. The multiplicity-two sphere is only the infinite-time limit.
Compactness: Compact
Surface genus: Not a surface
Topological ends: 0 (compact)
Entropy: Not applicable: Intrinsic MCF in a curved product manifold; Euclidean entropy / Gaussian area is not assigned.
Gaussian area: Not applicable: Intrinsic MCF in a curved product manifold; Euclidean entropy / Gaussian area is not assigned.

Ancient flows from unstable free-boundary minimal hypersurfaces

Source-informed schematic

Earliest recorded reference: 2026-02-09

Source construction and dimension restrictions are stated on the reading page. Illustrations have their own accuracy labels.

Surface case m = 2 · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Topology of the chosen two-sided free-boundary minimal hypersurface
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Genus of the chosen minimal surface.
Topological ends: No ends (compact parameter manifold).
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Hypersurface dimension m ≥ 3 · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Topology of the chosen two-sided free-boundary minimal hypersurface
Embeddedness: Embedded: The cited construction gives an embedding for this case.
Compactness: Compact
Surface genus: Not a smooth surface; surface genus is inapplicable.
Topological ends: No ends (compact parameter manifold).
Entropy: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.
Gaussian area: Not applicable: Euclidean Gaussian area and entropy are not assigned to this intrinsic curved-ambient flow.

Lagrangian sphere chains in Gibbons–Hawking spaces

Numerical quotient flow and circle-fiber diagram

Earliest recorded reference: 2026-06-27

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Circle-invariant Lagrangian sphere in a hyperkähler four-manifold · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Sphere S² at every finite time; limiting A_(N−1) chain has N−1 spheres, N ≥ 3.
Embeddedness: Embedded: The positive quotient graph lifts to an embedded sphere; junctions of the limiting chain occur only at infinite time.
Compactness: Compact
Surface genus: 0 for each finite-time sphere.
Topological ends: No ends (compact).
Entropy: Not applicable: Intrinsic mean curvature flow in a Riemannian ambient manifold; Euclidean Gaussian area and entropy are not assigned.
Gaussian area: Not applicable: Intrinsic mean curvature flow in a Riemannian ambient manifold; Euclidean Gaussian area and entropy are not assigned.

Immortal curve shortening with nonunique geodesic limits

Parameter-space schematic

Earliest recorded reference: 2026-08-05

Dimensions, topology, and embeddedness refer to the stated full ambient geometry and regular slices. See the source hypotheses and illustration conventions on this page.

Immortal closed curve in (S²,g) · Riemannian ambient manifold · Embedded

Ambient space: Riemannian ambient manifold
Topology: Circle S¹ at every finite time
Embeddedness: Embedded: The stated regular submanifold is embedded in its full ambient space.
Compactness: Compact
Surface genus: Curve, not surface.
Topological ends: No ends (compact).
Entropy: Not applicable: Intrinsic mean curvature flow in a Riemannian ambient manifold; Euclidean Gaussian area and entropy are not assigned.
Gaussian area: Not applicable: Intrinsic mean curvature flow in a Riemannian ambient manifold; Euclidean Gaussian area and entropy are not assigned.

Cite this page

BibTeX citation

@misc{mcf-stack-browse,
  author = {{MCF-stack: Special Solutions of Mean Curvature Flow contributors}},
  title = {Browse by properties},
  howpublished = {MCF-stack: Special Solutions of Mean Curvature Flow},
  url = {https://aosunmath.github.io/MCF-stack/browse/}
}

Comments

Comments are not enabled yet.