The round sphere minimizes entropy among closed self-shrinkers

TL;DR

This paper proves that among all closed self-shrinkers, the round sphere uniquely minimizes entropy, establishing it as the simplest singularity with a quantifiable gap to the next simplest.

Contribution

The authors demonstrate that the round sphere uniquely minimizes entropy among closed self-shrinkers and identify a positive gap to the second lowest entropy.

Findings

Round sphere has the lowest entropy among closed self-shrinkers.

There is a positive gap between the entropy of the sphere and the next lowest.

Entropy serves as a Lyapunov functional under mean curvature flow.

Abstract

The entropy of a hypersurface is a geometric invariant that measures complexity and is invariant under rigid motions and dilations. It is given by the supremum over all Gaussian integrals with varying centers and scales. It is monotone under mean curvature flow, thus giving a Lyapunov functional. Therefore, the entropy of the initial hypersurface bounds the entropy at all future singularities. We show here that not only does the round sphere have the lowest entropy of any closed singularity, but there is a gap to the second lowest.