Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere

TL;DR

This paper establishes a sharp quartic curvature pinching condition for the mean curvature flow in spheres, leading to convergence results and geometric estimates without relying on traditional integral methods.

Contribution

It generalizes Pu's convergence results by proving a new sharp quartic curvature pinching condition for mean curvature flow in spheres.

Findings

Submanifolds become approximately codimension one in high curvature regions.

The flow converges smoothly to a totally geodesic submanifold.

The approach avoids Stampacchia iteration and integral analysis.

Abstract

We prove a sharp quartic curvature pinching for the mean curvature flow in $\mathbb{S}^{n+m}$, $m\ge2$, which generalises Pu's work on the convergence of submanifolds in $\mathbb{S}^{n+m}$ to a round point. Using a blow up argument, we prove a codimension and a cylindrical estimate, where in regions of high curvature, the submanifold becomes approximately codimension one, quantitatively, and is weakly convex and moves by translation or is a self shrinker. With a decay estimate, the rescaling converges smoothly to a totally geodesic limit in infinite time, without using Stampacchia iteration or integral analysis.