  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 $M$.
