MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.13 Embedded rotational shrinking tori (Angenent, 1992; Colding–Minicozzi II, 2009; Drugan–Lee–X. H. Nguyen, 2017; Berchenko-Kogan, 2018)EB7B6435F145

Earliest recorded reference: (publication year).

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. Angenent — 1992 (publication year).
  2. Colding–Minicozzi II — 2009-08-26 (first arXiv submission).
  3. Drugan–Lee–X. H. Nguyen — 2017-08-30 (first arXiv submission).
  4. Berchenko-Kogan — 2018-08-24 (first arXiv submission).
EB7B6435F145

Angenent constructs an embedded rotational self-shrinker with topology $S^1\times S^{n-1}$ for each $n\geq 2$ in [Angenent1992]. We use the normalized profile equation from [DLN2018], Section 3.3 and equation (5.2), PDF pp. 8 and 14.

10374A017921

There is a simple closed profile $s\mapsto (x(s),r(s))$ in $r>0$ whose rotation gives

\[ X(s,\omega )=(x(s),r(s)\omega )\in \mathbb R\times \mathbb R^ n, \qquad \omega \in S^{n-1}(1). \]

For Euclidean arclength $s$ and tangent angle $\alpha $, the profile solves

\[ x'=\cos \alpha ,\qquad r'=\sin \alpha ,\qquad \alpha '=\left(\frac{n-1}{r}-\frac r2\right)\cos \alpha +\frac x2\sin \alpha . \]

Equivalently, it is a closed geodesic of $r^{2(n-1)}e^{-(x^2+r^2)/2}(dx^2+dr^2)$ on $r>0$. The theorem selects a closed orbit by shooting; the profile is not given by a closed formula. Its flow is $\sqrt{-t}\, \Sigma $, $t<0$.

  • The hypersurface is compact, embedded, complete and proper, with topology $S^1\times S^{n-1}$, rotational $O(n)$ symmetry and reflection symmetry across $x=0$. When $n=2$ it is a torus in $\mathbb R^3$.

  • It is not mean convex. Indeed, the compact embedded mean-convex classification in [CM12], Theorem 0.17, would force a sphere. It shrinks to the origin with a Type I singularity.

  • It is entropy unstable by [CM12], Theorem 0.12, which applies to this compact embedded non-spherical hypersurface. No exact entropy value or global uniqueness assertion is supplied here. Berchenko–Kogan's numerical calculation estimates its entropy as approximately $1.85122$ ([BerchenkoKogan2021], Section 5, PDF pp. 10–12). This estimate has no certified error enclosure; Section 6, Question 6.3, leaves rigorous error bounds open. The proved bound for the separately constructed variational torus is recorded in Example 2.28.

The interactive figure revolves a numerically closed solution of this ODE; its caption records the shooting residual. The profile is not circular. The variational torus in [DLN2018], Theorem 2, is a separate existence construction; that survey does not identify it with Angenent's torus.

Angenent's shrinking torus. A closed rotational shrinking profile in R3, with outer radius approximately 3.31470827 and inner radius 0.43712397. Return position and angle residuals before mesh rounding are below 1e-9. Numerical closure does not prove existence or uniqueness.
Numerical ODE surface. A closed rotational shrinking profile in R3, with outer radius approximately 3.31470827 and inner radius 0.43712397. Return position and angle residuals before mesh rounding are below 1e-9. Numerical closure does not prove existence or uniqueness. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy
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.

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References on this page

  1. [Angenent1992] Sigurd B. Angenent. Shrinking doughnuts. In Nonlinear Diffusion Equations and Their Equilibrium States, 3, volume 7 of Progress in Nonlinear Differential Equations and Their Applications, pages 21–38. Birkhäuser, Boston, MA, 1992.All uses in this project ↩ 1
  2. [DLN2018] Gregory Drugan, Hojoo Lee, and Xuan Hien Nguyen. A survey of closed self-shrinkers with symmetry. Results in Mathematics, 73(1):32, 2018. Article 32, 32 pages; locators refer to the supplied published edition. URL: https://arxiv.org/abs/1708.09113, doi:10.1007/s00025-018-0763-3.All uses in this project ↩ 1 ↩ 2
  3. [CM12] Tobias H. Colding and William P. Minicozzi, II. Generic mean curvature flow I: generic singularities. Annals of Mathematics, 175(2):755–833, 2012. URL: https://doi.org/10.4007/annals.2012.175.2.7, doi:10.4007/annals.2012.175.2.7.All uses in this project ↩ 1 ↩ 2
  4. [BerchenkoKogan2021] Yakov Berchenko-Kogan. The entropy of the Angenent torus is approximately 1.85122. Experimental Mathematics, 30(4):587–594, 2021. Source locators refer to arXiv version 2, 5 August 2019, 14 pages; numerical estimates, not certified error bounds. URL: https://arxiv.org/abs/1808.08163v2, doi:10.1080/10586458.2019.1583616.All uses in this project ↩ 1
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