Li-Yau Conformal Volume and Colding-Minicozzi Entropy of Self-Shrinkers

TL;DR

This paper establishes a lower bound on the Colding-Minicozzi entropy of self-shrinkers using the Li-Yau conformal volume, verifies a conjecture for certain topologies, and introduces new auxiliary functionals of independent interest.

Contribution

It introduces the concept of normalized Li-Yau conformal volume to bound entropy and verifies a conjecture for two-dimensional real projective plane self-shrinkers.

Findings

Conformal volume bounds entropy from below.

Bound is sharp for planes.

Verification of the entropy conjecture for specific topologies.

Abstract

We show the (normalized) Li-Yau conformal volume of a self-shrinker of mean curvature flow in Euclidean space bounds its Colding-Minicozzi entropy from below. This bound is independent of codimension and sharp on planes. As an application we verify a conjecture of Colding-Minicozzi about the entropy of closed self-shrinkers of arbitrary codimension for self-shrinkers that are topologically two-dimensional real projective planes. As part of the proof we introduce two auxiliary functionals which we call stable conformal volume and virtual entropy which should be of independent interest.