Mean curvature flow of graphs in warped products

TL;DR

This paper investigates the mean curvature flow of graphs in warped product manifolds, proving long-time existence, smoothness, and convergence under certain conditions, extending understanding of geometric flows in complex ambient spaces.

Contribution

It establishes the global existence and smoothness of the mean curvature flow of graphs in warped products, and analyzes conditions for convergence to a limit surface.

Findings

Flow exists for all time

Evolving hypersurfaces remain graphs

Flow converges under certain conditions

Abstract

Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $\varphi$ be a positive smooth function on $M$. In the warped product $M\times_\varphi\mathbb R$, we study the flow by the mean curvature of a locally Lipschitz continuous graph on $M$ and prove that the flow exists for all time and that the evolving hypersurface is $C^\infty$ for $t>0$ and is a graph for all $t$. Moreover, under certain conditions, the flow has a well defined limit.