MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.28 Variational embedded shrinking tori (Drugan–X. H. Nguyen, 2017; Chu–Sun, 2023)7E48D8C7D552

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. Drugan–X. H. Nguyen — 2017-08-29 (first arXiv submission).
  2. Chu–Sun — 2023-08-11 (first arXiv submission).
7E48D8C7D552

This is the variational construction in [DruganNguyen2018], Theorem 1, arXiv version 1, PDF p. 1. Its identification with the torus obtained by Angenent's shooting construction is not established in this source. The separate entry records the proved Gaussian-area bound, without claiming that the two constructions give distinct surfaces.

9F3C76F8391F

For every integer $n\geq 2$, the weighted half-plane $\{ (r,x):r>0\} $ with metric

\[ g_ n=r^{2(n-1)}e^{-(r^2+x^2)/2}(dr^2+dx^2) \]

contains a simple closed geodesic $\gamma _ n=(r,x)$ whose weighted length satisfies

\[ L_{g_ n}(\gamma _ n)<2\int _0^\infty s^{n-1}e^{-s^2/4}\, ds. \]

Revolving this curve gives a normalized self-shrinker

\[ X(u,\omega )=(r(u)\omega ,x(u)),\qquad (u,\omega )\in S^1\times S^{n-1}(1), \qquad \vec H=-X^\perp /2. \]

The profile is obtained by a variational and modified curve-flow argument, not by a closed formula. The smooth MCF is $\sqrt{-t}\, X$, $t<0$.

  • This is a compact embedded hypersurface of dimension $n$ in $\mathbb R^{n+1}$, with topology $S^1\times S^{n-1}$ and rotational $O(n)$ symmetry. It is complete, proper, boundaryless and has no ends. For $n=2$ it is an embedded genus-one torus in $\mathbb R^3$.

  • The rotational Gaussian integral and the length bound give

    \[ \lambda (\mathcal T_ n)=F_{0,1}(\mathcal T_ n) =(4\pi )^{-n/2}|S^{n-1}(1)|L_{g_ n}(\gamma _ n)<2. \]

    Thus this family has a proved entropy upper bound in every stated dimension. The theorem does not provide an exact entropy value.

  • In dimension two, this is the shrinking doughnut $\mathcal T$ used by [ChuSun2025], Theorem 1.3 and Corollary 1.7, to produce genus-one singularities and an embedded genus-one shrinker of strictly lower entropy. That conclusion does not identify the lower-entropy model with a particular previously constructed family.

  • The asserted bound is attached to this variationally constructed torus. A numerical value for the Angenent shooting profile is a different type of evidence and does not establish that the profiles coincide.

The illustration shows the weighted half-plane construction and a clearly labelled torus topology schematic. Reusing a numerical Angenent mesh as an exact model of this construction would assume the unproved identification.

Weighted half-plane construction of a closed geodesic; no identity with the numerical Angenent profile is assumed.
Variational construction schematic. Weighted half-plane construction of a closed geodesic; no identity with the numerical Angenent profile is assumed. Open drawing · Numerical references and credits
Ambient space, topology, embeddedness and entropy

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).

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References on this page

  1. [DruganNguyen2018] Gregory Drugan and Xuan Hien Nguyen. Shrinking doughnuts via variational methods. The Journal of Geometric Analysis, 28:3725–3746, 2018. Source locators refer to arXiv version 1, 29 August 2017, 18 pages. URL: https://arxiv.org/abs/1708.08808v1, doi:10.1007/s12220-017-9976-z.All uses in this project ↩ 1
  2. [ChuSun2025] Adrian Chun-Pong Chu and Ao Sun. Genus one singularities in mean curvature flow. Geometry & Topology, 29(8):4299–4340, 2025. Source locators refer to arXiv version 3, 6 March 2025, 37 pages. URL: https://arxiv.org/abs/2308.05923v3, doi:10.2140/gt.2025.29.4299.All uses in this project ↩ 1
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