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Transactions of the American Mathematical Society

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On almost-Fuchsian manifolds

Authors: Zheng Huang and Biao Wang
Journal: Trans. Amer. Math. Soc. 365 (2013), 4679-4698
MSC (2010): Primary 53A10; Secondary 53C12, 57M05
Published electronically: April 2, 2013
MathSciNet review: 3066768
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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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Additional Information

Zheng Huang
Affiliation: Department of Mathematics, The City University of New York, Staten Island, New York 10314
MR Author ID: 759027

Biao Wang
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
MR Author ID: 919266

Received by editor(s): May 18, 2010
Received by editor(s) in revised form: June 12, 2011
Published electronically: April 2, 2013
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.