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Minimal surfaces in finite volume noncompact hyperbolic $ 3$-manifolds


Authors: Pascal Collin, Laurent Hauswirth, Laurent Mazet and Harold Rosenberg
Journal: Trans. Amer. Math. Soc. 369 (2017), 4293-4309
MSC (2010): Primary 53A10
DOI: https://doi.org/10.1090/tran/6859
Published electronically: February 23, 2017
MathSciNet review: 3624410
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Abstract: We prove there exists a compact embedded minimal surface in a complete finite volume hyperbolic $ 3$-manifold $ \mathcal {N}$. We also obtain a least area, incompressible, properly embedded, finite topology, $ 2$-sided surface. We prove a properly embedded minimal surface of bounded curvature has finite topology. This determines its asymptotic behavior. Some rigidity theorems are obtained.


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Pascal Collin
Affiliation: Institut de mathematiques de Toulouse, Université Paul Sabatier, 118, route de Narbonne, F-31062 Toulouse cedex, France
Email: collin@math.ups-tlse.fr

Laurent Hauswirth
Affiliation: Université Paris-Est, LAMA (UMR 8050), UPEM, UPEC, CNRS, F-77454, Marne-la-Vallee, France
Email: hauswirth@univ-mlv.fr

Laurent Mazet
Affiliation: Université Paris-Est, LAMA (UMR 8050), UPEC, UPEM, CNRS, 61, avenue du Général de Gaulle, F-94010 Créteil cedex, France
Email: laurent.mazet@math.cnrs.fr

Harold Rosenberg
Affiliation: Instituto Nacional de Matematica Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320, Rio de Janeiro-RJ, Brazil
Email: rosen@impa.br

DOI: https://doi.org/10.1090/tran/6859
Received by editor(s): August 19, 2015
Received by editor(s) in revised form: September 19, 2015
Published electronically: February 23, 2017
Additional Notes: The authors were partially supported by grant ANR-11-IS01-0002.
Article copyright: © Copyright 2017 American Mathematical Society