Closed Embedded Self-shrinkers of Mean Curvature Flow

Abstract

In this article, we show the existence of closed embedded self-shrinkers in {mathbb {R}}^{n+1} that are topologically of type S^1times M, where Msubset 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^1times S^ktimes S^ksubset {mathbb {R}}^{2k+2} for any k. If the number of distinct principle curvatures of M is one, the resulting self-shrinker is topologically S^1times S^{n-1} and the construction recovers Angenent’s shrinking doughnut (Angenent in Shrinking doughnuts, Birkhäuser, Boston, pp 21–38).