The Extension for Mean Curvature Flow with Finite Integral Curvature in Riemannian Manifolds

TL;DR

This paper establishes conditions under which the mean curvature flow in Riemannian manifolds can be extended beyond finite time, based on finite total mean curvature, and demonstrates the optimality of these conditions.

Contribution

The paper proves that finite total mean curvature ensures the extension of mean curvature flow solutions and shows the optimality of this integral condition.

Findings

Flow can be extended if total mean curvature is finite.

The integral condition for extension is optimal.

Extension results apply to finite time intervals in Riemannian manifolds.

Abstract

We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature on a finite time interval $[0,T)$ can be extended over time $T$. Moreover, we show that the condition is optimal in some sense.