A note on the entropy of mean curvature flow

TL;DR

This paper investigates the entropy of hypersurfaces under mean curvature flow, establishing that low entropy implies the hypersurface is spherical or hyperplanar, thereby linking entropy levels to geometric shape classification.

Contribution

It proves that low entropy hypersurfaces under mean curvature flow are diffeomorphic to spheres or are hyperplanes, providing new geometric classification results based on entropy.

Findings

Low entropy hypersurfaces are diffeomorphic to spheres.

Self-shrinkers with low entropy are hyperplanes.

Entropy is non-increasing under mean curvature flow.

Abstract

The entropy of a hypersurface is given by the supremum over all F-functionals with varying centers and scales, and is invariant under rigid motions and dilations. As a consequence of Huisken's monotonicity formula, entropy is non-increasing under mean curvature flow. We show here that a compact mean convex hypersurface with some low entropy is diffeomorphic to a round sphere. We will also prove that a smooth self-shrinker with low entropy is exact a hyperplane.