Mean curvature flow of graphs in warped products

Abstract:

Let $M$ be a complete Riemannian manifold which is either 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.