Entire $\sigma_k$ curvature flow in Minkowski space

TL;DR

This paper investigates the evolution of noncompact spacelike hypersurfaces under the $\sigma_k$ curvature flow in Minkowski space, proving long-term existence and convergence to self-expanders under specific initial conditions.

Contribution

It establishes the long-time existence and convergence of the $\sigma_k$ curvature flow for noncompact hypersurfaces in Minkowski space, a novel result in this geometric flow context.

Findings

Flow exists for all time under certain initial conditions.

Rescaled flow converges to a self-expander.

Provides conditions ensuring long-term stability of the flow.

Abstract

In this paper, we study the $\sigma_k$ curvature flow of noncompact spacelike hypersurfaces in Minkowski space. We prove that if the initial hypersurface satisfies certain conditions, then the flow exists for all time. Moreover, we show that after rescaling, the flow converges to a self-expander.