Mean curvature flow of arbitrary codimension in complex projective spaces

TL;DR

This paper studies the evolution of submanifolds in complex projective spaces under mean curvature flow, showing convergence to round points or geodesic submanifolds under certain conditions, and establishing a new sphere theorem.

Contribution

It extends convergence results of mean curvature flow to arbitrary codimension in complex projective spaces and introduces a new differentiable sphere theorem.

Findings

Flow converges to round points or totally geodesic submanifolds

Provides a new differentiable sphere theorem

Improves previous convergence theorems

Abstract

In this paper, we investigate the mean curvature flow of submanifolds of arbitrary codimension in $\mathbb{C}\mathbb{P}^m$. We prove that if the initial submanifold satisfies a pinching condition, then the mean curvature flow converges to a round point in finite time, or converges to a totally geodesic submanifold as $t \rightarrow \infty$. Consequently, we obtain a new differentiable sphere theorem for submanifolds in $\mathbb{C}\mathbb{P}^m$. Our work improves the convergence theorem for mean curvature flow due to Pipoli and Sinestrari {\cite{PiSi2015}}.