On almost-Fuchsian manifolds

Abstract

An almost-Fuchsian manifold is a class of complete hyperbolic three-manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal curvatures everywhere in the range of ( − 1 , 1 ) (-1,1) . In such a manifold, the minimal surface is unique and embedded, hence one can parametrize these hyperbolic three-manifolds by their minimal surfaces. In this paper we obtain estimates on several geometric and analytical quantities of an almost-Fuchsian manifold M M in terms of the data on the minimal surface. In particular, we obtain an upper bound for the hyperbolic volume of the convex core of M M and an upper bound on the Hausdorff dimension of the limit set associated to M M . We also constructed a quasi-Fuchsian manifold which admits more than one minimal surface, and it does not admit a foliation of closed surfaces of constant mean curvature.