Deforming submanifolds of arbitrary codimension in a sphere

TL;DR

This paper establishes convergence results for the mean curvature flow of closed submanifolds in spheres under integral curvature conditions, leading to differentiable sphere theorems for these submanifolds.

Contribution

It provides new convergence theorems for mean curvature flow in spheres and derives differentiable sphere theorems for submanifolds under integral curvature conditions.

Findings

Convergence theorems for mean curvature flow in spheres.

Differentiable sphere theorems for certain submanifolds.

Results under integral curvature conditions.

Abstract

In this paper, we prove some convergence theorems for the mean curvature flow of closed submanifolds in the unit sphere $\mathbb{S}^{n+d}$ under integral curvature conditions. As a consequence, we obtain several differentiable sphere theorems for certain submanifolds in $\mathbb{S}^{n+d}$.