Uniqueness Theorem for non-compact mean curvature flow with possibly unbounded curvatures

TL;DR

This paper establishes new uniqueness and backward uniqueness theorems for non-compact mean curvature flows, including cases with unbounded curvatures, extending previous results and also applying similar methods to Ricci flows.

Contribution

It introduces generalized uniqueness theorems for mean curvature flow with unbounded curvatures and extends the approach to Ricci flows, broadening the scope of flow uniqueness results.

Findings

Proved uniqueness theorems for mean curvature flow with unbounded curvatures.

Established backward uniqueness for mean curvature flow with bounded curvatures.

Extended the energy method approach to Ricci flows.

Abstract

In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqueness result on Ricci flows. A backward uniqueness theorem is also proved for mean curvature flow with bounded curvatures.