Sharp Entropy Bounds for Self-Shrinkers in Mean Curvature Flow

TL;DR

This paper establishes lower bounds on the entropy of smooth, closed self-shrinkers in mean curvature flow, showing that the entropy is minimized by round spheres, which are characterized by equality cases.

Contribution

It proves sharp entropy bounds for self-shrinkers, linking topological complexity to geometric entropy and characterizing equality cases as round spheres.

Findings

Entropy of self-shrinkers is bounded below by that of round spheres.

Equality in entropy bounds characterizes round spheres.

Results connect topology with geometric flow properties.

Abstract

Let $M\subset {\mathbf R}^{m+1}$ be a smooth, closed, codimension-one self-shrinker (for mean curvature flow) with nontrivial $k^{\rm th}$ homology. We show that the entropy of $M$ is greater than or equal to the entropy of a round $k$-sphere, and that if equality holds, then $M$ is a round $k$-sphere in ${\mathbf R}^{k+1}$.