Bernstein theorem for translating solitons of hypersurfaces

TL;DR

This paper establishes a Bernstein-type theorem and a gap theorem for translating solitons of hypersurfaces in Euclidean space, providing conditions under which these solitons are hyperplanes, based on monotonicity formulas and curvature norms.

Contribution

It introduces new Bernstein and gap theorems for translating solitons, extending understanding of their geometric properties and conditions for flatness.

Findings

Proves a monotonicity formula for translating solitons.

Establishes conditions under which solitons are hyperplanes.

Shows that small $L^n$ norm of second fundamental form implies flatness.

Abstract

In this paper, we prove a monotonicity formula and some Bernstein type results for translating solitons of hypersurfaces in $\re^{n+1}$, giving some conditions under which a trantranslating soliton is a hyperplane. We also show a gap theorem for the translating soliton of hypersurfaces in $R^{n+k}$, namely, if the $L^n$ norm of the second fundamental form of the soliton is small enough, then it is a hyperplane.