On Almost-Fuchsian Manifolds

Details On Almost-Fuchsian Manifolds On Almost-Fuchsian Manifolds

2009-07-16

arxiv.org/abs/0907.2899v4

Mathematics

Differential Geometry

19 pages, 2 figures

Scientific paper

Title is changed. We reorganize this project and this is the first part where we focus on the geometry and dynamics of the almost-Fuchsian manifolds. We obtain estimates on several geometric and analytical quantities of an almost-Fuchsian manifold in terms of the data on its (unique) minimal surface. In particular, we obtain an upper bound for the hyperbolic volume of its convex core, and an upper bound on the Hausdor? dimension of the limit set associated to this almost-Fuchsian manifold. 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.

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