Mean curvature flow in Fuchsian manifolds

TL;DR

This paper studies the mean curvature flow in Fuchsian hyperbolic 3-manifolds, showing that certain initial surfaces evolve smoothly and converge to a totally geodesic surface, providing new examples of such flows in Riemannian manifolds.

Contribution

It demonstrates the long-time existence and convergence of mean curvature flow for a class of initial surfaces in Fuchsian manifolds, a novel result in this context.

Findings

Flow exists for all time for certain initial surfaces.

Flow converges to the totally geodesic surface.

Provides calculations for warped product settings.

Abstract

Motivated by questions in detecting minimal surfaces in hyperbolic manifolds, we study the behavior of geometric flows in complete hyperbolic three-manifolds. In most cases the flows develop singularities in finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic 3-manifolds (Fuchsian manifolds) which are warped products of a closed surface of genus at least two and the real line. In particular, we prove that there exists a large class of closed initial surfaces, as geodesic graphs over the totally geodesic surface $\Sigma$, such that the mean curvature flow exists for all time and converges to $\Sigma$. This is among the first examples of converging mean curvature flows of compact hypersurfaces in Riemannian manifolds. We also provide some useful calculations for the general warped product setting.