Surfaces pinched by normal curvature for mean curvature flow in space forms

TL;DR

This paper studies the evolution of compact surfaces in four-dimensional space forms under mean curvature flow, establishing convergence results when certain normal curvature pinching conditions are met, extending previous theorems.

Contribution

It generalizes Baker-Nguyen's convergence theorem by proving new convergence results under normal curvature pinching conditions in 4D space forms.

Findings

Convergence theorems for mean curvature flow under normal curvature pinching.

Extension of previous convergence results to higher-dimensional space forms.

Conditions ensuring smooth convergence of evolving surfaces.

Abstract

In this paper, we investigate the mean curvature flow of compact surfaces in $4$-dimensional space forms. We prove the convergence theorems for the mean curvature flow under certain pinching conditions involving the normal curvature, which generalise Baker-Nguyen's convergence theorem.