Gauss map of translating solitons of mean curvature flow

TL;DR

This paper investigates Bernstein-type theorems for translating solitons in mean curvature flow, focusing on the Gauss map's image containment, leading to classical results in minimal submanifold theory.

Contribution

It establishes new Bernstein-type theorems for translating solitons based on Gauss map image constraints, extending classical minimal surface results.

Findings

Gauss map images contained in compact subsets imply rigidity of translating solitons

Special case yields classical Bernstein's theorem for minimal submanifolds

Provides conditions under which translating solitons are affine

Abstract

In this short note we study Bernstein's type theorem of translating solitons whose images of their Gauss maps are contained in compact subsets in an open hemisphere of the standard $\mathbf{S}^n$ (see Theorem 1.1). As a special case we get a classical Bernstein's type theorem in minimal submanifolds in $\mathbf{R}^{n+1}$ (see Corollary 1.2).