Monotonicity of Perelman $\mathcal{W}$-Entropy of Mean Curvature Flow
Xiang-Dong Li, Qi Yan
TL;DR
The paper redefines Perelman’s W-entropy for mean curvature flow by working with evolving domains in R^{n+1} and a forward heat equation coupled to a gradient flow. It derives a monotonicity formula for the new W-entropy under weakly convex mean curvature flow, leveraging Hamilton’s Harnack inequality for the flow. A rigidity result shows that equality in the monotonicity occurs only for shrinking spheres with Gaussian heat kernel, linking the entropy to a canonical self-similar solution. This work extends the entropy framework from Ricci flow to mean curvature flow and clarifies when entropy nonincreasing behavior characterizes the most symmetric evolutions.
Abstract
In this paper, we study Perelman' s $ \mathcal{W}$ entropy for mean curvature flow in $\mathbb{R}^{n+1}$. Analogously to Perelman's $\mathcal{W}$-entropy defined for Ricci flow, K. Ecker in \cite{Ecker07} defined a functional $\mathcal{W}$ for the mean curvature flow in $\mathbb{R}^{n+1}$ and the region it encloses, and made the conjecture that this functional is monotonically increasing in time. We modify K. Ecker's definition and, using Hamilton's Harnack inequality for mean curvature flow, prove that our redefined $\mathcal{W}$-entropy is monotonically decreasing in time. Additionally, we provide a rigidity theorem for this $\mathcal{W}$-entropy.