Stability property and Dirichlet problem for translating solitons

TL;DR

This paper investigates the stability, mean curvature properties, and Dirichlet problem solutions for translating solitons in higher-dimensional Euclidean spaces, providing new insights into their geometric behavior and stability conditions.

Contribution

It establishes that the infimum of the mean curvature is zero for translating solitons and explores stability criteria and volume growth conditions, along with solving the Dirichlet problem in higher codimensions.

Findings

Infimum of mean curvature is zero for translating solitons.

Stability conditions for complete hypersurface translating solitons.

Weighted volume exhibits exponential growth when mean curvature norm is less than one.

Abstract

In this paper, we prove that the infimum of the mean curvature is zero for a translating solitons of hypersurface in $\re^{n+k}$. We give some conditions under which a complete hypersurface translating soliton is stable. We show that if the norm of its mean curvature is less than one, then the weighted volume may have exponent growth. We also study the Dirichlet problem for graphic translating solitons in higher codimensions.