The Gauss image of $\lambda$-hypersurfaces and a Bernstein type problem

TL;DR

This paper investigates rigidity properties of mbda-hypersurfaces in Euclidean space via their Gauss maps, proving that entire graphical mbda-hypersurfaces must be hyperplanes, thus addressing a Bernstein type problem.

Contribution

It establishes a Bernstein type theorem for mbda-hypersurfaces, showing that entire graphic solutions are necessarily hyperplanes, which is a new rigidity result.

Findings

Entire graphic mbda-hypersurfaces are hyperplanes.

Rigidity theorems for mbda-hypersurfaces under Gauss map.

Addresses Bernstein type problem for mbda-hypersurfaces.

Abstract

In this paper, our purpose is to study rigidity theorems for $\lambda$-hypersurfaces in Euclidean space under Gauss map. As a Bernstein type problem for $\lambda$-hypersurfaces, we prove that an entirely graphic $\lambda$-hypersurface in Euclidean space is a hyperplane.