Harnack Inequality for $f$-Mean Curvature Flow

TL;DR

The paper extends Li-Yau-Hamilton type Harnack inequalities to the $f$-mean curvature flow in Euclidean space, where the flow is the gradient flow of the weighted area with density $e^{-f}$. By deriving evolution equations for geometric data and introducing a Harnack quantity $Z$ built from $H_f$, its derivatives, and auxiliary fields, the authors prove a differential Harnack inequality under the condition $\overline{\nabla}^3 f \equiv 0$, using commutation relations and a maximum principle. They further establish an integral Harnack inequality, yielding curvature-control bounds such as $|h|^2/H_f^2 \le C^2$ and a ratio bound for $H_f$ along space-time paths, with explicit forms that recover Hamilton’s classical results for mean curvature flow in suitable limits. These results illuminate how weighting by $e^{-f}$ interacts with curvature-driven flow, providing tools to control geometric quantities along the weighted flow and linking gradient flow of the weighted area to established MCF theory.

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 $e^{-f}$.

Theorems & Definitions (21)

• Theorem 1.1: Hamilton 95'
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Lemma 3.1
• proof
• Lemma 3.2
• Lemma 3.3
• proof