Rigidity of spacelike translating solitons in pseudo-Euclidean space

TL;DR

This paper classifies complete spacelike translating solitons in pseudo-Euclidean space, proving they are all spacelike m-planes, and extends rigidity results to higher codimension solitons with applications to Gauss images.

Contribution

It provides a classification of complete spacelike translating solitons and generalizes rigidity theorems to higher codimension cases.

Findings

Complete spacelike translating solitons are spacelike m-planes.

Nonexistence of nontrivial complete spacelike translating solitons.

Extended rigidity theorems with applications to Gauss images.

Abstract

In this paper, we investigate the parametric version and non-parametric version of rigidity theorem of spacelike translating solitons in pseudo-Euclidean space $\mathbb{R}^{m+n}_{n}$. Firstly, we classify $m$-dimensional complete spacelike translating solitons in $\mathbb{R}^{m+n}_{n}$ by affine technique and classical gradient estimates, and prove the only complete spacelike translating solitons in $\mathbb{R}^{m+n}_{n}$ are the spacelike $m$-planes. This result provides another proof of a nonexistence theorem for complete spacelike translating solitons in \cite{C-Q}, and a simple proof of rigidity theorem in \cite{X-H}. Secondly, we generalize the rigidity theorem of entire spacelike Lagrangian translating solitons in \cite{X-Z} to spacelike translating solitons with general codimensions. As a directly application of theorem, we obtain two interesting corollaries in terms of Gauss image.