Translating solutions and the entire Hessian curvature flow in Minkowski space

TL;DR

This paper investigates the evolution of noncompact spacelike hypersurfaces in Minkowski space under the k-Hessian curvature flow, establishing existence, convergence to translating solutions, and asymptotic behaviors.

Contribution

It introduces new results on the existence and convergence of solutions to the k-Hessian curvature flow in Minkowski space, including the construction of translating solutions.

Findings

Existence of translating solutions with prescribed asymptotics.

Global existence and convergence of the normalized flow.

Hypersurfaces evolve towards translating solutions over time.

Abstract

In this paper, we study the $k$-Hessian curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We first prove the existence of translating solutions with given asymptotic behavior. Then, we prove that for strictly convex initial hypersurface satisfying certain conditions, the curvature flow exists for all time, and the normalized flow converges to a translating solution.