Mean curvature flow in Fuchsian manifolds

Abstract

Motivated by the goal of detecting minimal surfaces in hyperbolic manifolds, we study geometric flows in complete hyperbolic [Formula: see text]-manifolds. In general, the flows might develop singularities at some finite time. In this paper, we investigate the mean curvature flow in a class of complete hyperbolic [Formula: see text]-manifolds (Fuchsian manifolds) which are warped products of a closed surface of genus at least two and [Formula: see text]. We show that for a large class of closed initial surfaces, which are graphs over the totally geodesic surface [Formula: see text], the mean curvature flow exists for all time and converges to [Formula: see text]. This is among the first examples of converging mean curvature flows starting from closed hypersurfaces in Riemannian manifolds. We also provide calculations for the general warped product setting which will be useful for further works.