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Closed minimal surfaces in cusped hyperbolic three-manifolds

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Abstract

Motivated by classical theorems on minimal surface theory in compact hyperbolic 3-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic 3-manifold of finite volume. We prove any closed immersed incompressible surface can be deformed to a closed immersed least area surface within its homotopy class in any cusped hyperbolic 3-manifold. Our techniques highlight how special structures of these cusped hyperbolic 3-manifolds prevent any least area minimal surface going too deep into the cusped region.

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Acknowledgements

We would like to thank Richard Canary, Joseph Maher and Alan Reid for helpful discussions. We also thank the support from PSC-CUNY research awards. Z. H. acknowledges supports from U.S. NSF Grants DMS 1107452, 1107263, 1107367 “RNMS: Geometric Structures and Representation varieties” (the GEAR Network) and a Grant from the Simons Foundation (#359635, Zheng Huang). It was a pleasure to discuss some aspects of this project at Intensive Period on Teichmüller theory and 3-manifold at Centro De Giorgi, Pisa, Italy, and Workshop on Minimal Surfaces and Hyperbolic Geometry at IMPA, Rio, Brazil. We thank the referee for careful reading and helpful suggestions.

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Authors and Affiliations

  1. Department of Mathematics, The City University of New York, Staten Island, NY, 10314, USA

    Zheng Huang

  2. The Graduate Center, The City University of New York, 365 Fifth Ave., New York, NY, 10016, USA

    Zheng Huang

  3. Department of Mathematics and Computer Science, Queensborough Community College, The City University of New York, 222-05 56th Avenue Bayside, New York, NY, 11364, USA

    Biao Wang

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  1. Zheng Huang
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  2. Biao Wang
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Correspondence to Zheng Huang.

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Huang, Z., Wang, B. Closed minimal surfaces in cusped hyperbolic three-manifolds. Geom Dedicata 189, 17–37 (2017). https://doi.org/10.1007/s10711-016-0215-8

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  • DOI: https://doi.org/10.1007/s10711-016-0215-8

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