A Bernstein type theorem for minimal hypersurfaces via Gauss maps

TL;DR

This paper proves that certain minimal hypersurfaces in Euclidean space must be hyperplanes if their Gauss maps avoid a specific region, extending Bernstein-type theorems to higher dimensions.

Contribution

It establishes a new Bernstein-type theorem for minimal hypersurfaces based on the behavior of their Gauss maps.

Findings

Minimal hypersurfaces with Gauss maps avoiding a neighborhood of a half-equator are hyperplanes.

The result applies to complete, embedded minimal hypersurfaces with Euclidean volume growth.

Provides conditions under which minimal hypersurfaces are necessarily flat.

Abstract

Let $M$ be an $n$-dimensional smooth oriented complete embedded minimal hypersurface in $\mathbb{R}^{n+1}$ with Euclidean volume growth. We show that if the image under the Gauss map of $M$ avoids some neighborhood of a half-equator, then $M$ must be an affine hyperplane.