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Cylindrically bounded constant mean curvature surfaces in $ \mathbb{H} ^2\times\mathbb{R}$


Author: Laurent Mazet
Journal: Trans. Amer. Math. Soc. 367 (2015), 5329-5354
MSC (2010): Primary 53A10
DOI: https://doi.org/10.1090/S0002-9947-2015-06171-6
Published electronically: April 2, 2015
MathSciNet review: 3347174
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper it is proved that a properly embedded constant mean curvature surface in $ \mathbb{H}^2\times \mathbb{R}$ which has finite topology and stays at a finite distance from a vertical geodesic line is invariant by rotation around a vertical geodesic line.


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  • [1] Juan A. Aledo, José M. Espinar, and José A. Gálvez, Height estimates for surfaces with positive constant mean curvature in $ \mathbb{M}^2\times \mathbb{R}$, Illinois J. Math. 52 (2008), no. 1, 203-211. MR 2507241 (2010e:53006)
  • [2] Sébastien Cartier and Laurent Hauswirth, Deformations of constant mean curvature-$ 1/2$ surfaces in $ \Bbb H^2\times \Bbb R$ with vertical ends at infinity, Comm. Anal. Geom. 22 (2014), no. 1, 109-148. MR 3194376, https://doi.org/10.4310/CAG.2014.v22.n1.a2
  • [3] Pascal Collin, Topologie et courbure des surfaces minimales proprement plongées de $ \mathbf {R}^3$, Ann. of Math. (2) 145 (1997), no. 1, 1-31 (French). MR 1432035 (98d:53010), https://doi.org/10.2307/2951822
  • [4] P. Collin and R. Krust, Le problème de Dirichlet pour l'équation des surfaces minimales sur des domaines non bornés, Bull. Soc. Math. France 119 (1991), no. 4, 443-462 (French, with English summary). MR 1136846 (92m:53007)
  • [5] José M. Espinar, José A. Gálvez, and Harold Rosenberg, Complete surfaces with positive extrinsic curvature in product spaces, Comment. Math. Helv. 84 (2009), no. 2, 351-386. MR 2495798 (2010c:53086), https://doi.org/10.4171/CMH/165
  • [6] David Hoffman, Jorge H. S. de Lira, and Harold Rosenberg, Constant mean curvature surfaces in $ M^2\times \mathbf {R}$, Trans. Amer. Math. Soc. 358 (2006), no. 2, 491-507. MR 2177028 (2006e:53016), https://doi.org/10.1090/S0002-9947-05-04084-5
  • [7] Wu-Teh Hsiang and Wu-Yi Hsiang, On the uniqueness of isoperimetric solutions and imbedded soap bubbles in noncompact symmetric spaces. I, Invent. Math. 98 (1989), no. 1, 39-58. MR 1010154 (90h:53078), https://doi.org/10.1007/BF01388843
  • [8] Nicholas J. Korevaar, Rob Kusner, William H. Meeks III, and Bruce Solomon, Constant mean curvature surfaces in hyperbolic space, Amer. J. Math. 114 (1992), no. 1, 1-43. MR 1147718 (92k:53116), https://doi.org/10.2307/2374738
  • [9] Nicholas J. Korevaar, Rob Kusner, and Bruce Solomon, The structure of complete embedded surfaces with constant mean curvature, J. Differential Geom. 30 (1989), no. 2, 465-503. MR 1010168 (90g:53011)
  • [10] R. Kusner, R. Mazzeo, and D. Pollack, The moduli space of complete embedded constant mean curvature surfaces, Geom. Funct. Anal. 6 (1996), no. 1, 120-137. MR 1371233 (97b:58022), https://doi.org/10.1007/BF02246769
  • [11] Olga A. Ladyzhenskaya and Nina N. Uraltseva, Linear and quasilinear elliptic equations, Translated from the Russian by Scripta Technica, Inc. Translation editor: Leon Ehrenpreis, Academic Press, New York, 1968. MR 0244627 (39 #5941)
  • [12] Laurent Mazet, A general halfspace theorem for constant mean curvature surfaces, Amer. J. Math. 135 (2013), no. 3, 801-834. MR 3068403, https://doi.org/10.1353/ajm.2013.0027
  • [13] William H. Meeks III, The topology and geometry of embedded surfaces of constant mean curvature, J. Differential Geom. 27 (1988), no. 3, 539-552. MR 940118 (89h:53025)
  • [14] Barbara Nelli and Harold Rosenberg, Global properties of constant mean curvature surfaces in $ \mathbb{H}^2\times \mathbb{R}$, Pacific J. Math. 226 (2006), no. 1, 137-152. MR 2247859 (2007i:53010), https://doi.org/10.2140/pjm.2006.226.137
  • [15] Barbara Nelli and Harold Rosenberg, Simply connected constant mean curvature surfaces in $ {\mathbb{H}}^2\times \mathbb{R}$, Michigan Math. J. 54 (2006), no. 3, 537-543. MR 2280494 (2008f:53007), https://doi.org/10.1307/mmj/1163789914
  • [16] Renato H. L. Pedrosa and Manuel Ritoré, Isoperimetric domains in the Riemannian product of a circle with a simply connected space form and applications to free boundary problems, Indiana Univ. Math. J. 48 (1999), no. 4, 1357-1394. MR 1757077 (2001k:53120), https://doi.org/10.1512/iumj.1999.48.1614
  • [17] Joaquín Pérez and Antonio Ros, Properly embedded minimal surfaces with finite total curvature, The global theory of minimal surfaces in flat spaces (Martina Franca, 1999), Lecture Notes in Math., vol. 1775, Springer, Berlin, 2002, pp. 15-66. MR 1901613, https://doi.org/10.1007/978-3-540-45609-4_2
  • [18] Harold Rosenberg, Some recent developments in the theory of minimal surfaces in 3-manifolds, Publicações Matemáticas do IMPA. [IMPA Mathematical Publications], Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2003. 24$ ^{\rm o}$ Colóquio Brasileiro de Matemática. [24th Brazilian Mathematics Colloquium]. MR 2028922 (2005b:53015)
  • [19] J. Serrin, The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables, Philos. Trans. Roy. Soc. London Ser. A 264 (1969), 413-496. MR 0282058 (43 #7772)
  • [20] Joel Spruck, Interior gradient estimates and existence theorems for constant mean curvature graphs in $ M^n\times \mathbf {R}$, Pure Appl. Math. Q. 3 (2007), no. 3, Special Issue: In honor of Leon Simon., 785-800. MR 2351645 (2009b:58025)

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Additional Information

Laurent Mazet
Affiliation: Laboratoire d’Analyse et Mathématiques Appliquées, Université Paris-Est, CNRS UMR8050, UFR des Sciences et Technologie, Bâtiment P3 4eme étage, 61 avenue du Général de Gaulle, 94010 Créteil cedex, France
Email: laurent.mazet@math.cnrs.fr

DOI: https://doi.org/10.1090/S0002-9947-2015-06171-6
Received by editor(s): January 22, 2013
Published electronically: April 2, 2015
Article copyright: © Copyright 2015 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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