Cylindrical Estimates for High Codimension Mean Curvature Flow

TL;DR

This paper investigates high codimension mean curvature flow under a quadratic curvature condition, establishing gradient estimates and demonstrating that singularities are quantitatively cylindrical, extending concepts of convexity to higher codimensions.

Contribution

It introduces a new quadratic curvature condition for high codimension submanifolds and proves gradient estimates and cylindrical behavior near singularities.

Findings

Established a pointwise gradient estimate for the flow.

Proved that singularities are quantitatively cylindrical.

Extended convexity notions to high codimension settings.

Abstract

We study high codimension mean curvature flow of a submanifold $\mathcal{M}^n$ of dimension $n$ in Euclidean space $\mathbb{R}^{n+k}$ subject to the quadratic curvature condition $ |A|^{2}\leq c_n |H|^{2}, c _n = \min\{ \frac{4}{3n} , \frac{1}{n-2}\}$. This condition extends the notion of two-convexity for hypersurfaces to high codimension submanifolds. We analyse singularity formation in the mean curvature flow of high codimension by directly proving a pointwise gradient estimate. We then show that near a singularity the surface is quantitatively cylindrical.