Counting minimal surfaces in quasi-Fuchsian three-manifolds
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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.References
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Additional Information
- 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
- MR Author ID: 759027
- 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
- MR Author ID: 919266
- Email: bwang@wesleyan.edu, biwang@qcc.cuny.edu
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 6063-6083
- MSC (2010): Primary 53A10; Secondary 57M05
- DOI: https://doi.org/10.1090/tran/6172
- MathSciNet review: 3356929