Closed Embedded Self-shrinkers of Mean Curvature Flow

Abstract

In this article, we show the existence of closed embedded self-shrinkers in Rn+1 that are topologically of type S1×M, where M⊂Sn is any isoparametric hypersurface in Sn for which the multiplicities of the principle curvatures agree. This yields new examples of closed self-shrinkers, for example self-shrinkers of topological type S1×Sk×Sk⊂R2k+2 for any k. If the number of distinct principle curvatures of M is one, the resulting self-shrinker is topologically S1×Sn−1 and the construction recovers Angenent’s shrinking doughnut (Angenent in Shrinking doughnuts, Birkhäuser, Boston, pp 21–38).