Harnack Inequality for $f$-Mean Curvature Flow

TL;DR

This paper establishes a Harnack inequality for the $f$-mean curvature flow, a gradient flow of a weighted area functional, extending classical estimates to a new geometric flow setting.

Contribution

It proves a Li-Yau-Hamilton type Harnack estimate for the $f$-mean curvature flow in Euclidean space, a novel result in geometric analysis.

Findings

Derived a Harnack inequality for the $f$-mean curvature flow.

Extended classical Harnack estimates to weighted geometric flows.

Provided new tools for analyzing gradient flows of weighted functionals.

Abstract

In this paper, we prove a Li-Yau-Hamilton type Harnack estimate for the $f$-mean curvature flow in Euclidean space, which can be viewed as a gradient flow of the weighed area functional with the measure density function $e^{-f}$.