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

TL;DR

This paper proves a topological property of certain self-shrinkers with low entropy, showing the link of their asymptotic cone separates the sphere into two parts, leading to a uniqueness result for the sphere as the entropy minimizer.

Contribution

It establishes a topological separation property for asymptotically conical self-shrinkers with low entropy, confirming a conjecture about the sphere's uniqueness in dimension two.

Findings

The link of the asymptotic cone separates the sphere into two connected components.

The sphere uniquely minimizes entropy among non-flat two-dimensional self-shrinkers.

Confirmed a conjecture of Colding-Ilmanen-Minicozzi-White in dimension two.

Abstract

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