Mean Curvature Flows in Almost Fuchsian Manifolds

An almost Fuchsian manifold is a quasi-Fuchsian hyperbolic three-manifold that contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1,1). In such a hyperbolic three-manifold, the minimal surface is unique and embedded, hence one can parametrize these three-manifolds by their minimal surfaces. We prove that any closed surface which is a graph over any fixed surface of small principal curvatures can be deformed into the minimal surface via the mean curvature flow. We also obtain an upper bound for the hyperbolic volume of the convex core of M, as well as estimates of the Hausdorff dimension of the limit set for $M$.