Sharp entropy bounds for self-shrinkers in mean curvature flow

Abstract

Let $M\subset {ℝ}^{m+1}$ be a smooth, closed, codimension-one self-shrinker (for mean curvature flow) with nontrivial $k^{\text{ th}}$ 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 ${ℝ}^{k+1}$.