A remark on a Bernstein type result for $\lambda$-hypersurfaces

TL;DR

This paper proves that complete hypersurfaces in Euclidean space with Gauss images in an open hemisphere are proper and extends Bernstein-type results to $mbda$-hypersurfaces, addressing a problem posed by Cheng-Wei.

Contribution

It establishes properness of certain hypersurfaces and provides a Bernstein-type theorem for $mbda$-hypersurfaces, advancing understanding of their geometric properties.

Findings

Complete hypersurfaces with Gauss image in an open hemisphere are proper.

A Bernstein-type theorem for $mbda$-hypersurfaces is proved.

Addresses a conjecture by Cheng-Wei on Bernstein problems.

Abstract

We proved that any complete hypersurface in the Euclidean space $\mathbb{R}^{n+1}$ whose Gauss image is contained in an open hemisphere has to be proper. As applications, we derive a counterpart of Hoffman-Osserman-Schoen's result for $\lambda$-hypersurfaces, which gives an affirmative answer to the Bernstein type problem proposed by Cheng-Wei \cite{CW14b}.