Translating graphs by Mean curvature flow in $\M^n\times\Real$

TL;DR

This paper investigates the evolution of graphs in product manifolds under mean curvature flow, proving convergence to translating surfaces and establishing non-existence results in negatively curved surfaces.

Contribution

It introduces new results on the behavior of mean curvature flow for graphs in product manifolds, including convergence and non-existence theorems.

Findings

Solutions converge to translating surfaces in $\M^n imes eal$.

No complete vertically translating graphs exist in $\M^2 imes eal$ with negative Gaussian curvature.

Provides conditions for the existence and non-existence of certain translating solutions.

Abstract

In this work, we study graphs in $\M^n\times\Real$ that are evolving by the mean curvature flow over a bounded domain on $\M^n$, with prescribed contact angle in the boundary. We prove that solutions converge to translating surfaces in $\M^n\times\Real$. Also, for a Riemannian manifold $\M^2$ with negative Gaussian curvature at each point, we show non-existence of complete vertically translating graphs in $\M^2\times\Real$.