MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.21 Nguyen's one-ended shrinking saddlesEDE40E8E1379

Earliest recorded reference: (first arXiv submission).

Paper credits and chronology

Paper credits are chronological and include every author of the cited papers. They identify sources, not exclusive discovery credit. Background-only citations do not determine the date of this example. A classical example may predate the references recorded here.

  1. X. H. Nguyen — 2011-06-26 (first arXiv submission).
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For every sufficiently large integer $m$, Nguyen constructs a smooth complete embedded surface $\widetilde M_ m\subset \mathbb R^3$ of genus $m-1$, with one asymptotically conical end. The source uses

\[ \mathbf H_{\widetilde M_ m}=-\widetilde X^\perp . \]

Thus the catalogue's normalized example is $M_ m=\sqrt{2}\, \widetilde M_ m$, which satisfies $\mathbf H=-X^\perp /2$, and its flow is $M_ m(t)=\sqrt{-t}\, M_ m$ for $t<0$.

The construction replaces the intersection circle of a sphere and a plane by a bent, scaled singly periodic Scherk surface, then solves the nonlinear shrinker equation by a perturbation argument. In the catalogue normalization the input sphere has radius $2$. This is an existence construction, with no closed parametrization of the completed shrinker. Its proved properties and construction data include the following.

  • The surface has dimension two and codimension one. It is connected, orientable, noncompact, proper, and without boundary. Outside a compact set it is a graph over the exterior of a planar disk. Consequently it has one topological end; the asymptotic cone is not an assertion that the end is planar.

  • The $m$ periods of the Scherk piece produce genus $m-1$. The existence theorem requires $m$ sufficiently large and does not establish the small-$m$ cases.

  • The surface is invariant under a half-turn about the $x$-axis and under reflections in vertical planes making angles $\pi /(2m)+j\pi /m$, $j\in \mathbb Z$, with that axis. These are finite symmetries, not full rotational symmetry.

  • As $m\to \infty $, the source proves convergence to the spherical input on compact subsets of the upper half-ball in Theorem 1(iv), and convergence to the singly periodic Scherk model after translating and rescaling the joining region in Theorem 1(vi). These limits describe different spatial scales.

  • The conical graphical end gives quadratic area growth and finite entropy. In the stated normalization $F_{0,1}(M_ m)=\lambda (M_ m)$; no scalar entropy value or uniform numerical bound is recorded here. The theorem does not give a global uniqueness or stability classification.

Source: [NguyenShrinker2014], Theorem 1 and its preceding topology paragraph, PDF p. 2; the construction of the outer graphical piece and Proposition 14, PDF p. 13; Definition 21, PDF p. 20; the final fixed-point argument, PDF pp. 24–25. This is another sphere–plane desingularization construction beside Example 2.22; no identification or noncongruence between the two families is asserted. The interactive construction schematic shows a connected saddle belt, with a cutaway exposing the inner disk and lower cap. Its unwrapped view triangulates the exact minimal ingredient

\[ \Sigma _0=\{ (x,y,z)\in \mathbb R^3:\sin y=\sinh x\, \sinh z\} , \]

which has period $2\pi $ in the $y$-direction (Section 2, PDF p. 7). The four wings attach to the two caps, the inner disk, and the exterior sheet (Definition 13, PDF p. 13). The global drawing uses an illustrative wrapping and cap map, with a freely chosen conical profile; it does not compute the source's unbalancing parameter, matched rotational caps, or nonlinear correction. The displayed period counts are not asserted to lie in the existence range.

Connected Nguyen construction schematic with a periodic saddle belt, a cutaway exposing the inner disk and lower cap, and an unwrapped Scherk ingredient. Not a computed self-shrinker.
Scherk-based construction schematic. A connected sphere–plane construction model with two caps, an inner disk, and one outer sheet joined through a Scherk saddle belt. Rotate it, open the cutaway, or inspect the unwrapped Scherk minimal surface. The displayed period counts illustrate genus m − 1; the existence theorem requires sufficiently large m. The global shape and conical profile are schematic, not a solved shrinker. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

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.

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References on this page

  1. [NguyenShrinker2014] Xuan Hien Nguyen. Construction of complete embedded self-similar surfaces under mean curvature flow. part III. Duke Mathematical Journal, 163(11):2023–2056, 2014. Citations refer to arXiv v3, July 2, 2014, 26 PDF pages. The source uses H-vector = -X-perp; multiply space by sqrt(2) for the catalogue normalization. URL: https://arxiv.org/abs/1106.5272v3, doi:10.1215/00127094-2795108.All uses in this project ↩ 1
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