Compactness and rigidity of self-shrinking surfaces

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.