Geometry & Topology

The Dirichlet Problem for constant mean curvature graphs in $\mathbb{M}\times\mathbb{R}$

Abigail Folha and Harold Rosenberg

Full-text: Open access

Abstract

We study graphs of constant mean curvature H>0 in M× for M a Hadamard surface, ie a complete simply connected surface with curvature bounded above by a negative constant a. We find necessary and sufficient conditions for the existence of these graphs over bounded domains in M, having prescribed boundary data, possibly infinite.

Article information

Source
Geom. Topol., Volume 16, Number 2 (2012), 1171-1203.

Dates
Received: 21 February 2011
Revised: 5 March 2012
Accepted: 10 April 2012
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513732416

Digital Object Identifier
doi:10.2140/gt.2012.16.1171

Mathematical Reviews number (MathSciNet)
MR2946806

Zentralblatt MATH identifier
1281.53013

Subjects
Primary: 53A10: Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42]
Secondary: 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

Keywords
Hadamard surface constant mean curvature Dirichlet problem

Citation

Folha, Abigail; Rosenberg, Harold. The Dirichlet Problem for constant mean curvature graphs in $\mathbb{M}\times\mathbb{R}$. Geom. Topol. 16 (2012), no. 2, 1171--1203. doi:10.2140/gt.2012.16.1171. https://projecteuclid.org/euclid.gt/1513732416


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References

  • P Collin, H Rosenberg, Construction of harmonic diffeomorphisms and minimal graphs, Ann. of Math. 172 (2010) 1879–1906
  • A Folha, S Melo, The Dirichlet problem for constant mean curvature graphs in $\mathbb{H}\times\mathbb{R}$ over unbounded domains, Pacific J. Math. 251 (2011) 37–65
  • J A Gálvez, H Rosenberg, Minimal surfaces and harmonic diffeomorphisms from the complex plane onto certain Hadamard surfaces, Amer. J. Math. 132 (2010) 1249–1273
  • D Gilbarg, N S Trudinger, Elliptic partial differential equations of second order, second edition, Grundl. Math. Wissen. 224, Springer, Berlin (1983)
  • L Hauswirth, H Rosenberg, J Spruck, Infinite boundary value problems for constant mean curvature graphs in $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$, Amer. J. Math. 131 (2009) 195–226
  • H Jenkins, J Serrin, Variational problems of minimal surface type. I, Arch. Rational mech. Anal. 12 (1963) 185–212
  • H Jenkins, J Serrin, Variational problems of minimal surface type II. Boundary value problems for the minimal surface equation, Arch. Rational Mech. Anal. 21 (1966) 321–342
  • F Labourie, Un lemme de Morse pour les surfaces convexes, Invent. Math. 141 (2000) 239–297
  • Y Li, L Nirenberg, Regularity of the distance function to the boundary, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 29 (2005) 257–264
  • L Mazet, M M Rodríguez, H Rosenberg, The Dirichlet problem for the minimal surface equation, with possible infinite boundary data, over domains in a Riemannian surface, Proc. Lond. Math. Soc. 102 (2011) 985–1023
  • B Nelli, H Rosenberg, Minimal surfaces in $\mathbb{H}^2\times\mathbb{R}$, Bull. Braz. Math. Soc. 33 (2002) 263–292
  • A L Pinheiro, A Jenkins–Serrin theorem in $M^2\times\mathbb{R}$, Bull. Braz. Math. Soc. 40 (2009) 117–148
  • H Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993) 211–239
  • H Rosenberg, Minimal surfaces in $\mathbb{M}^2\times\mathbb{R}$, Illinois J. Math. 46 (2002) 1177–1195
  • H Rosenberg, R Souam, E Toubiana, General curvature estimates for stable $H$–surfaces in $3$–manifolds and applications, J. Differential Geom. 84 (2010) 623–648
  • J Spruck, Infinite boundary value problems for surfaces of constant mean curvature, Arch. Rational Mech. Anal. 49 (1972) 1–31
  • J Spruck, Interior gradient estimates and existence theorems for constant mean curvature graphs in $M^n\times R$, Pure Appl. Math. Q. 3 (2007) 785–800