Convexity Estimates for High Codimension Mean Curvature Flow

TL;DR

This paper studies the evolution of high codimension submanifolds under mean curvature flow, proving they become asymptotically convex and converge to convex translating solutions at singularities.

Contribution

It establishes convexity estimates for high codimension mean curvature flow and shows convergence to convex translating solutions at singularities.

Findings

Submanifolds become asymptotically convex during flow.

At singularities, rescaled flows converge to convex translating solutions.

Eigenvalue of second fundamental form grows slower than mean curvature.

Abstract

We consider the evolution by mean curvature of smooth $n$-dimensional submanifolds in $\mathbb{R}^{n+k}$ which are compact and quadratically pinched. We will be primarily interested in flows of high codimension, the case $k\geq 2$. We prove that our submanifold is asymptotically convex, that is the first eigenvalue of the second fundamental form in the principal mean curvature direction blows up at a strictly slower rate than the mean curvature vector. We use this convexity estimate to show that at a singular time of the flow, there exists a rescaling that converges to a smooth codimension-one limiting flow which is convex and moves by translation.