Three Bernstein type theorems for hypersurfaces with zero Gaussian curvature

TL;DR

This paper establishes Bernstein type theorems characterizing entire convex hypersurfaces with zero Gaussian curvature as hyperplanes under certain conditions, in both Euclidean and Minkowski spaces.

Contribution

It extends Bernstein type results to convex hypersurfaces with zero Gaussian curvature, including new conditions involving mean curvature and Minkowski space considerations.

Findings

Zero Gaussian curvature convex hypersurfaces are hyperplanes if mean curvature vanishes at infinity.

Counterexamples show zero Gaussian convex spacelike hypersurfaces are not necessarily hyperplanes without extra conditions.

Similar results are obtained for hypersurfaces in Minkowski space without timelike points.

Abstract

In this paper, we prove Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in both Euclidean and Minkowski context. A supplementary example illustrates that zero Gaussian convex spacelike hypersurfaces are not necessary hyperplanes without additional conditions. We show that a zero Gaussian curvature convex hypersurface must be a hyperplane if the mean curvature goes to zero at infinity. In the Minkowski context, we prove similar results for hypersurface without timelike points.