Sharp Entropy Bounds for Self-Shrinkers in Mean Curvature Flow

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