Monotonicity of Perelman $\mathcal{W}$-Entropy of Mean Curvature Flow

TL;DR

This paper investigates a modified Perelman $\\mathcal{W}$-entropy for mean curvature flow, proving its monotonic decrease over time and establishing a rigidity theorem, thus extending entropy concepts from Ricci flow to mean curvature flow.

Contribution

The paper introduces a modified $\\mathcal{W}$-entropy for mean curvature flow and proves its monotonic decreasing property using Hamilton's Harnack inequality.

Findings

The redefined $\\mathcal{W}$-entropy is monotonically decreasing in time.

A rigidity theorem related to the $\\mathcal{W}$-entropy is established.

The approach extends entropy methods from Ricci flow to mean curvature flow.

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.