Mean Curvature Flows of Closed Hypersurfaces in Warped Product Manifolds

TL;DR

This paper studies the evolution of closed hypersurfaces under mean curvature flow within warped product manifolds, showing long-term existence and convergence to a totally geodesic hypersurface under certain conditions.

Contribution

It establishes conditions ensuring the global existence and convergence of mean curvature flows for hypersurfaces in warped product manifolds, extending previous results to a broader class of ambient spaces.

Findings

Flow exists for all time under specified conditions.

Flow converges to a totally geodesic hypersurface.

Results apply to a large class of initial hypersurfaces.

Abstract

We investigate the mean curvature flows in a class of warped product manifolds with closed hypersurfaces fibering over $\mathbb{R}$. In particular, we prove that under natural conditions on the warping function and Ricci curvature bound for the ambient space, there exists a large class of closed initial hypersurfaces, as geodesic graphs over the totally geodesic hypersurface $\Sigma$, such that the mean curvature flow starting from $S_0$ exists for all time and converges to $\Sigma$.