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Counting minimal surfaces in quasi-Fuchsian three-manifolds


Authors: Zheng Huang and Biao Wang
Journal: Trans. Amer. Math. Soc. 367 (2015), 6063-6083
MSC (2010): Primary 53A10; Secondary 57M05
DOI: https://doi.org/10.1090/tran/6172
Published electronically: April 9, 2015
MathSciNet review: 3356929
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Abstract: It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many stable ones. In this paper, for any prescribed integer $ N > 0$, we construct a quasi-Fuchsian manifold which contains at least $ 2^N$ such minimal surfaces. As a consequence, there exists some simple closed Jordan curve on $ S_{\infty }^2$ such that there are at least $ 2^N$ disk-type complete minimal surfaces in $ \mathbb{H}^3$ sharing this Jordan curve as the asymptotic boundary.


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Zheng Huang
Affiliation: Department of Mathematics, The City University of New York, Staten Island, New York 10314; The Graduate Center, The City University of New York, 365 Fifth Avenue, New York, New York 10016
Email: zheng.huang@csi.cuny.edu

Biao Wang
Affiliation: Department of Mathematics, Wesleyan University, Middletown, Connecticut 06459
Address at time of publication: Department of Mathematics and Computer Science, Queensborough Community College, City University of New York, 222-05 56th Avenue, Bayside, New York 11364
Email: bwang@wesleyan.edu, biwang@qcc.cuny.edu

DOI: https://doi.org/10.1090/tran/6172
Received by editor(s): September 10, 2012
Received by editor(s) in revised form: February 24, 2013, and April 10, 2013
Published electronically: April 9, 2015
Article copyright: © Copyright 2015 American Mathematical Society