Entire downward translating solitons to the mean curvature flow in Minkowski space

TL;DR

This paper investigates entire translating solutions to mean curvature flow in Minkowski space, revealing their structure as convex solitons, their asymptotic behavior, and the nonuniqueness of solutions under perturbations.

Contribution

It characterizes the structure and asymptotics of entire translating solutions in Minkowski space, including convexity and blowdown limits, and demonstrates nonuniqueness through perturbations.

Findings

Solutions reduce to convex rank k solitons in Minkowski space.

Blowdown limits converge to positively homogeneous convex functions.

Existence of nonuniqueness via smooth perturbations of symmetric solutions.

Abstract

In this paper, we study entire translating solutions $u(x)$ to a mean curvature flow equation in Minkowski space. We show that if $\Sigma=\{(x, u(x))| x\in\mathbb{R}^n\}$ is a strictly spacelike hypersurface, then $\Sigma$ reduces to a strictly convex rank k soliton in $\mathbb{R}^{k, 1}$ (after splitting off trivial factors) whose "blowdown" converges to a multiple $\lambda\in(0, 1)$ of a positively homogeneous degree one convex function in $\mathbb{R}^k$. We also show that there is nonuniqueness as the rotationally symmetric solution may be perturbed to a solution by an arbitrary smooth order one perturbation.