MCF-stack: Special Solutions of Mean Curvature Flow

§ 2.22 One-ended high-genus shrinkers from sphere–plane gluing (Kapouleas–Kleene–Møller, 2011; Sun–Zhichao Wang, 2018)2285B0C987A6

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. Kapouleas–Kleene–Møller — 2011-06-27 (first arXiv submission).
  2. Sun–Zhichao Wang — 2018-11-13 (first arXiv submission).
2285B0C987A6

The original construction is [KKM2018], Theorem 1.1, arXiv version 3, PDF p. 2. It is among the examples cited in the introduction of [SunWang2020].

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There is an integer $g_0$ such that for every integer $g\geq g_0$ there exists a smooth complete embedded orientable self-shrinker $\Sigma _ g\subset \mathbb R^3$ of genus $g$, with

\[ \vec H=-\frac12X^\perp ,\qquad M_ t=\sqrt{-t}\, \Sigma _ g,\quad -\infty <t<0. \]

The construction desingularizes the intersection circle of a plane through the origin and $S^2(2)$. It is a gluing existence theorem; no explicit parametrization of $\Sigma _ g$ is provided.

  • The surface has dimension two and codimension one, is noncompact, has exactly one end, and separates $\mathbb R^3$ into two connected components. Its topology is an orientable closed genus-$g$ surface with one point removed. The compact core and graphical end give properness.

  • It is invariant under the dihedral symmetry group of order $4g+4$ specified in the construction.

  • Outside a ball the end is a graph over a plane, asymptotic to a nonplanar regular cone whose link is a nonzero smooth vertical graph over a great circle of $S^2$. The end has asymptotic height $u(\rho ,\theta )=\rho \sigma (\theta )+O(\rho ^{-1})$; the function $\sigma $ is determined by the construction, not an explicit formula. See equation (1.2), PDF p. 4.

  • As $g\to \infty $, the surfaces converge locally in Hausdorff distance to $S^2(2)\cup P$. The source gives an $O(R/g)$ Hausdorff bound inside $B_ R(0)$; convergence is locally smooth away from the intersection circle. The limiting union is singular, and is not one of the smooth shrinkers.

  • The conical geometry gives quadratic area growth and finite entropy. The source supplies no exact entropy value or numerical entropy bound for individual $g$. It records entropy instability on PDF p. 4. No uniqueness or identification with another construction is asserted.

The picture resolves the joining circle using a Scherk-based construction model. The local ingredient in the paper is the singly periodic minimal surface

\[ \sinh u\, \sinh v=\sin w,\qquad (u,v,w)\in \mathbb R^3, \]

with period $2\pi $ in $w$; see equation (5.1), PDF p. 9. Section 5 curls $k=g+1$ periods around the equator and attaches the four wings to the upper cap, lower cap, inner disk, and outer sheet. The genus relation is in Remark 5.12, PDF p. 12. The illustration uses a simplified coordinate map and an adjustable conical deformation, not the paper's matched initial surface or the nonlinear correction that solves the shrinker equation. Its cutaway exposes the inner disk and the joining belt. The displayed values $k=6,9,12$ illustrate the topology and do not assert that these genera lie in the theorem's existence range. The outer edge is a display truncation; the waviness is not the computed function $\sigma $. Figure 1 of the paper, PDF p. 3, is itself captioned as a conjectural genus-eight shrinker.

Connected Scherk-based sphere–plane construction model with a ring of saddle-shaped openings, spherical caps, an inner disk and one wavy outer sheet. Not a computed self-shrinker.
Scherk-based construction schematic. A connected model joins two sphere-like caps, an inner disk, and one outer sheet through a periodic Scherk belt. Rotate or use Cutaway to inspect the joining region. The period count and conical waviness are illustrative; this is not a numerically solved KKM shrinker. The theorem applies only to sufficiently large genus, with no explicit threshold. The outer edge is a display truncation. The conjectural picture in the source’s Figure 1 is credited to Tom Ilmanen. Open drawing · Numerical references and credits

Ambient space, topology, embeddedness and entropy

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.

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References on this page

  1. [KKM2018] Nikolaos Kapouleas, Stephen J. Kleene, and Niels Martin Møller. Mean curvature self-shrinkers of high genus: non-compact examples. Journal für die reine und angewandte Mathematik, 739:1–39, 2018. Source locators refer to arXiv version 3, 18 November 2014, 41 pages. URL: https://arxiv.org/abs/1106.5454v3, doi:10.1515/crelle-2015-0050.All uses in this project ↩ 1
  2. [SunWang2020] Ao Sun and Zhichao Wang. Compactness of self-shrinkers in $\mathbb R^3$ with fixed genus. Advances in Mathematics, 367:107110, 2020. Article 107110; source locators refer to arXiv version 2, 27 November 2019, 33 pages. URL: https://arxiv.org/abs/1811.05387v2, doi:10.1016/j.aim.2020.107110.All uses in this project ↩ 1
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