A sharp convergence theorem for the mean curvature flow in spheres I

TL;DR

This paper establishes a precise convergence theorem for the mean curvature flow of submanifolds in spheres, enhancing previous results and leading to a new differentiable sphere theorem.

Contribution

It provides a sharper convergence theorem for mean curvature flow in spheres, improving Baker's theorem and introducing a new differentiable sphere theorem.

Findings

Proved a sharp convergence theorem for mean curvature flow in spheres.

Improved upon Baker's convergence theorem.

Derived a new differentiable sphere theorem for submanifolds.

Abstract

In this paper, we prove a sharp convergence theorem for the mean curvature flow of arbitrary codimension in spheres which improves Baker's convergence theorem. In particular, we obtain a new differentiable sphere theorem for submanifolds in spheres.