A Bernstein theorem for complete spacelike constant mean curvature hypersurfaces in Minkowski space

TL;DR

This paper establishes a Bernstein-type theorem for complete spacelike hypersurfaces with constant mean curvature in Minkowski space, showing they are linear if their Gauss map image is bounded on one side.

Contribution

It extends previous Bernstein theorems by removing the boundedness assumption on the Gauss map image for spacelike hypersurfaces in Minkowski space.

Findings

Gradient estimate for Gauss maps of hypersurfaces

Proved that hypersurfaces with bounded Gauss map image are linear

Extended Bernstein theorems to hypersurfaces with parallel mean curvature vector

Abstract

We obtain a gradient estimate for the Gauss maps from complete spacelike constant mean curvature hypersurfaces in Minkowski space into the hyperbolic space. As applications, we prove a Bernstein theorem which says that if the image of the Gauss map is bounded from one side, then the spacelike constant mean curvature hypersurface must be linear. This result extends the previous theorems obtained by B. Palmer and Y.L. Xin where they assume that the image of the Gauss map is bounded. We also proved a Bernstein theorem for spacelike complete surfaces with parallel mean curvature vector in four-dimensional spaces.