Geometry & Topology

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

Abigail Folha and Harold Rosenberg

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Abstract

We study graphs of constant mean curvature H>0 in M×ℝ 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
    Mathematical Reviews (MathSciNet): MR2726102
    Zentralblatt MATH: 1209.53010
    Digital Object Identifier: doi:10.4007/annals.2010.172.1879
  • 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
    Mathematical Reviews (MathSciNet): MR2794614
    Digital Object Identifier: doi:10.2140/pjm.2011.251.37
  • 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
    Mathematical Reviews (MathSciNet): MR2732346
    Digital Object Identifier: doi:10.1353/ajm.2010.0001
  • D Gilbarg, N S Trudinger, Elliptic partial differential equations of second order, second edition, Grundl. Math. Wissen. 224, Springer, Berlin (1983)
    Mathematical Reviews (MathSciNet): MR737190
    Zentralblatt MATH: 0562.35001
  • 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
    Mathematical Reviews (MathSciNet): MR2488489
    Digital Object Identifier: doi:10.1353/ajm.0.0040
  • H Jenkins, J Serrin, Variational problems of minimal surface type. I, Arch. Rational mech. Anal. 12 (1963) 185–212
    Mathematical Reviews (MathSciNet): MR0145194
    Zentralblatt MATH: 0122.39602
    Digital Object Identifier: doi:10.1007/BF00281225
  • 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
    Mathematical Reviews (MathSciNet): MR0190811
    Zentralblatt MATH: 0171.08301
    Digital Object Identifier: doi:10.1007/BF00282252
  • F Labourie, Un lemme de Morse pour les surfaces convexes, Invent. Math. 141 (2000) 239–297
    Mathematical Reviews (MathSciNet): MR1775215
    Zentralblatt MATH: 0981.52002
    Digital Object Identifier: doi:10.1007/s002220000069
  • Y Li, L Nirenberg, Regularity of the distance function to the boundary, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. 29 (2005) 257–264
    Mathematical Reviews (MathSciNet): MR2305073
  • 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
    Mathematical Reviews (MathSciNet): MR2806098
    Digital Object Identifier: doi:10.1112/plms/pdq032
  • B Nelli, H Rosenberg, Minimal surfaces in $\mathbb{H}^2\times\mathbb{R}$, Bull. Braz. Math. Soc. 33 (2002) 263–292
    Mathematical Reviews (MathSciNet): MR1940353
    Zentralblatt MATH: 1038.53011
    Digital Object Identifier: doi:10.1007/s005740200013
  • A L Pinheiro, A Jenkins–Serrin theorem in $M^2\times\mathbb{R}$, Bull. Braz. Math. Soc. 40 (2009) 117–148
    Mathematical Reviews (MathSciNet): MR2496117
    Zentralblatt MATH: 1176.53019
    Digital Object Identifier: doi:10.1007/s00574-009-0007-y
  • H Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993) 211–239
    Mathematical Reviews (MathSciNet): MR1216008
    Zentralblatt MATH: 0787.53046
  • H Rosenberg, Minimal surfaces in $\mathbb{M}^2\times\mathbb{R}$, Illinois J. Math. 46 (2002) 1177–1195
    Mathematical Reviews (MathSciNet): MR1988257
    Zentralblatt MATH: 1036.53008
    Project Euclid: euclid.ijm/1258138473
  • 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
    Mathematical Reviews (MathSciNet): MR2669367
    Digital Object Identifier: doi:10.4310/jdg/1279114303
    Project Euclid: euclid.jdg/1279114303
  • J Spruck, Infinite boundary value problems for surfaces of constant mean curvature, Arch. Rational Mech. Anal. 49 (1972) 1–31
    Mathematical Reviews (MathSciNet): MR0334010
    Zentralblatt MATH: 0263.53008
    Digital Object Identifier: doi:10.1007/BF00281471
  • 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
    Mathematical Reviews (MathSciNet): MR2351645
    Digital Object Identifier: doi:10.4310/PAMQ.2007.v3.n3.a6

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Geometry & Topology

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