On almost-Fuchsian manifolds

Huang, Zheng; Wang, Biao

doi:10.1090/S0002-9947-2013-05749-2

On almost-Fuchsian manifolds
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by Zheng Huang and Biao Wang PDF

Trans. Amer. Math. Soc. 365 (2013), 4679-4698 Request permission

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 $(-1,1)$. 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 $M$ 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 $M$ and an upper bound on the Hausdorff dimension of the limit set associated to $M$. 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.

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