Mean curvature flows of graphs sliding off to infinity in warped product manifolds

TL;DR

This paper investigates the behavior of mean curvature flows of graphs in warped product manifolds, demonstrating preservation of geodesic graphs and long-term convergence to points at infinity under certain conditions.

Contribution

It establishes conditions under which mean curvature flows preserve geodesic graphs and proves long-time existence and convergence for specific warping functions.

Findings

Flow preserves geodesic graphs for certain warping functions.

Long-time existence of the flow when warping functions tend to zero at infinity.

Curvature and derivatives decay to zero along the flow.

Abstract

We study mean curvature flows in a warped product manifold defined by a closed Riemannian manifold and $\mathbb{R}$. In such a warped product manifold, we can define the notion of a graph, called a geodesic graph. We prove that the curve shortening flow preserves a geodesic graph for any warping function, and the mean curvature flow of hypersurfaces preserves a geodesic graph for some monotone convex warping functions. In particular, we consider some warping functions that go to zero at infinity, which means that the curves or hypersurfaces go to a point at infinity along the flow. In such a case, we prove the long-time existence of the flow and that the curvature and its higher-order derivatives go to zero along the flow.