New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theory

TL;DR

This survey reviews recent advances in mean curvature flow of arbitrary codimension, highlighting new convergence and sphere theorems inspired by Yau rigidity theory, including optimal results for submanifolds with positive Ricci curvature.

Contribution

It introduces the first optimal convergence theorem for mean curvature flow in arbitrary codimension and derives new differentiable sphere theorems based on these results.

Findings

Optimal convergence theorem for mean curvature flow in arbitrary codimension

New differentiable sphere theorems for submanifolds

First optimal theorem for submanifolds with positive Ricci curvature

Abstract

In this survey, we will focus on the mean curvature flow theory with sphere theorems, and discuss the recent developments on the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the Yau rigidity theory of submanifolds. Several new differentiable sphere theorems for submanifolds are obtained as consequences of the convergence theorems for the mean curvature flow. It should be emphasized that Theorem 4.1 is an optimal convergence theorem for the mean curvature flow of arbitrary codimension, which implies the first optimal differentiable sphere theorem for submanifolds with positive Ricci curvature. Finally, we present a list of unsolved problems in this area.