High codimension mean curvature flow of spacelike-convex submanifolds with one spacelike codimension

TL;DR

This paper studies the mean curvature flow of spacelike submanifolds in pseudo-Euclidean space, proving curvature preservation and convergence to a point, extending classical results to higher codimension and indefinite metrics.

Contribution

It establishes curvature pinching and noncollapsing preservation, and proves that such submanifolds shrink to points, generalizing Huisken and Gage-Hamilton theorems to this setting.

Findings

Curvature pinching is preserved under the flow.

Submanifolds shrink to a point in finite time.

Flow solutions become asymptotic to shrinking spheres.

Abstract

In the pseudo-Euclidean space $\mathbb{R}^{n+1,k}$, we consider the mean curvature flow of $n$-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension $k$. We show that if the initial submanifold is compact and spacelike-convex (the acceleration along every geodesic is strictly spacelike), then natural quantities measuring curvature pinching and noncollapsing are preserved under the flow. Moreover, we prove an analogue of the Huisken and Gage-Hamilton theorems in this setting, which states that the mean curvature flow deforms any such submanifold to a point in finite time, and that the solution is asymptotic to a shrinking sphere in a maximally spacelike affine subspace $\mathbb{R}^{n+1,0}\subset \mathbb{R}^{n+1,k}$.