Entire spacelike hypersurfaces with constant $\sigma_k$ curvature in Minkowski space

TL;DR

This paper establishes existence and non-existence results for smooth, entire, convex spacelike hypersurfaces with constant $\sigma_k$ curvature in Minkowski space, and constructs examples with prescribed Gauss map images.

Contribution

It proves the existence of such hypersurfaces with prescribed lightlike directions and generalizes previous results, including constructing examples with bounded principal curvature.

Findings

Existence of hypersurfaces with prescribed Gauss map image.

Non-existence of certain entire convex hypersurfaces.

Construction of hypersurfaces with bounded principal curvature.

Abstract

In this paper, we prove the existence of smooth, entire, strictly convex, spacelike, constant $\sigma_k$ curvature hypersurfaces with prescribed lightlike directions in Minkowski space. This is equivalent to prove the existence of smooth, entire, strictly convex, spacelike, constant $\sigma_k$ curvature hypersurfaces with prescribed Gauss map image. We also show that there doesn't exist any entire, convex, strictly spacelike, constant $\sigma_k$ curvature hypersurfaces. Moreover, we generalize the result in \cite{RWX} and construct strictly convex, spacelike, constant $\sigma_k$ curvature hypersurface with bounded principal curvature, whose image of the Gauss map is the unit ball.