Closed embedded self-shrinkers of mean curvature flow

Oskar Riedler

Paper · Jul 11, 2022

Closed embedded self-shrinkers of mean curvature flow

Oskar Riedler

math.DGWider flowsarXiv:2207.04851v2

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Abstract

In this article we show the existence of closed embedded self-shrinkers in $R^{n + 1}$ that are topologically of type $S^{1} \times M$ , where $M \subset S^{n}$ is any isoparametric hypersurface in $S^{n}$ for which the multiplicities of the principle curvatures agree. This yields new examples of closed self-shrinkers, for example self-shrinkers of topological type $S^{1} \times S^{k} \times S^{k} \subset R^{2 k + 2}$ for any $k$ . If the number of distinct principle curvatures of $M$ is one the resulting self-shrinker is topologically $S^{1} \times S^{n - 1}$ and the construction recovers Angenent's shrinking doughnut.

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Submitted: Jul 11, 2022
Last revised: Apr 3, 2023 (v2)
Subjects: math.DG
Journal: Journal of Geometric Analysis 33, 172 (2023)
DOI: 10.1007/s12220-023-01217-w

Comments: 31 pages, 3 figures

Mean curvature flow Self-shrinkers & translators

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

In the guidebook

6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

[1] 6A.10
Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives
Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.

Closed embedded self-shrinkers of mean curvature flow

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