Closed embedded self-shrinkers of mean curvature flow

TL;DR

This paper constructs new examples of closed embedded self-shrinkers in Euclidean space with specific topologies, expanding the known classes of solutions to the mean curvature flow and including generalizations of Angenent's shrinking doughnut.

Contribution

It demonstrates the existence of closed embedded self-shrinkers with topologies derived from isoparametric hypersurfaces, broadening the known family of self-shrinkers in mean curvature flow.

Findings

New examples of closed self-shrinkers with topologies like $S^1\times M$

Construction includes self-shrinkers of type $S^1\times S^k\times S^k$

Recovers Angenent's shrinking doughnut for specific cases

Abstract

In this article we show the existence of closed embedded self-shrinkers in $\Bbb{R}^{n+1}$ 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 $S^1\times S^k\times S^k\subset \Bbb R^{2k+2}$ 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.