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

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.