Compactness and rigidity of self-shrinking surfaces

2021/08/09 by Lee, Tang-Kai
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2108.03919

Abstract

The entropy functional introduced by Colding and Minicozzi plays a fundamental role in the analysis of mean curvature flow. However, unlike the hypersurface case, relatively little about the entropy is known in the higher-codimension case. In this note, we use measure-theoretical techniques and rigidity results for self-shrinkers to prove a compactness theorem for a family of self-shrinking surfaces with low entropy. Based on this, we prove the existence of entropy minimizers among self-shrinking surfaces and improve some rigidity results.

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