Sharp Quartic Pinching for the Mean Curvature Flow in the Sphere

2024/08/15 by Vogiatzi, Artemis A.
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2408.08022

Abstract

We prove a sharp quartic curvature pinching for the mean curvature flow in \mathbbS^n+m, m≥2, which generalises Pu's work on the convergence of submanifolds in \mathbbS^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.

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