Cylindrically bounded constant mean curvature surfaces in ℍ²×ℝ

Mazet, Laurent

doi:10.1090/S0002-9947-2015-06171-6

Cylindrically bounded constant mean curvature surfaces in $\mathbb {H} ^2\times \mathbb {R}$
HTML articles powered by AMS MathViewer

by Laurent Mazet PDF

Trans. Amer. Math. Soc. 367 (2015), 5329-5354 Request permission

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.

References

Juan A. Aledo, José M. Espinar, and José A. Gálvez, Height estimates for surfaces with positive constant mean curvature in $\Bbb M^2\times \Bbb R$, Illinois J. Math. 52 (2008), no. 1, 203–211. MR 2507241
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, DOI 10.4310/CAG.2014.v22.n1.a2
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, DOI 10.2307/2951822
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
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, DOI 10.4171/CMH/165
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, DOI 10.1090/S0002-9947-05-04084-5
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, DOI 10.1007/BF01388843
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, DOI 10.2307/2374738
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
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, DOI 10.1007/BF02246769
Olga A. Ladyzhenskaya and Nina N. Ural’tseva, Linear and quasilinear elliptic equations, Academic Press, New York-London, 1968. Translated from the Russian by Scripta Technica, Inc; Translation editor: Leon Ehrenpreis. MR 0244627
Laurent Mazet, A general halfspace theorem for constant mean curvature surfaces, Amer. J. Math. 135 (2013), no. 3, 801–834. MR 3068403, DOI 10.1353/ajm.2013.0027
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
Barbara Nelli and Harold Rosenberg, Global properties of constant mean curvature surfaces in $\Bbb H^2\times \Bbb R$, Pacific J. Math. 226 (2006), no. 1, 137–152. MR 2247859, DOI 10.2140/pjm.2006.226.137
Barbara Nelli and Harold Rosenberg, Simply connected constant mean curvature surfaces in ${\Bbb H}^2\times \Bbb R$, Michigan Math. J. 54 (2006), no. 3, 537–543. MR 2280494, DOI 10.1307/mmj/1163789914
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, DOI 10.1512/iumj.1999.48.1614
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, DOI 10.1007/978-3-540-45609-4_{2}
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$^\textrm {o}$ Colóquio Brasileiro de Matemática. [24th Brazilian Mathematics Colloquium]. MR 2028922
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 282058, DOI 10.1098/rsta.1969.0033
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, DOI 10.4310/PAMQ.2007.v3.n3.a6