A Topological Property of Asymptotically Conical Self-Shrinkers of Small Entropy

摘要

For any asymptotically conical self-shrinker with entropy less than or equalto that of a cylinder we show that the link of the asymptotic cone mustseparate the unit sphere into exactly two connected components, bothdiffeomorphic to the self-shrinker. Combining this with recent work of Brendle,we conclude that the round sphere uniquely minimizes the entropy among allnon-flat two-dimensional self-shrinkers. This confirms a conjecture ofColding-Ilmanen-Minicozzi-White in dimension two.