Closed Embedded Self-shrinkers of Mean Curvature Flow

Abstract

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