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The non-linear Plateau problem in non-positively curved manifolds
Author:
Graham Smith
Journal:
Trans. Amer. Math. Soc. 365 (2013), 1109-1124
MSC (2010):
Primary 58E12; Secondary 35J25, 35J60, 53A10, 53C21, 53C42
Posted:
October 31, 2012
Full-text PDF
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Additional Information
Abstract: Using the Perron method, we prove the existence of hypersurfaces of prescribed special Lagrangian curvature with prescribed boundary inside complete Riemannian manifolds of non-positive curvature.
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Andersson L., Barbot T., Béguin F., Zeghib A., Cosmological time versus CMC time in spacetimes of constant curvature, arXiv:math/0701452
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Alexander, Mohammad
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Wong, Topology of Riemannian submanifolds with prescribed
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(2011d:53059), http://dx.doi.org/10.1215/00127094-2010-018
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Guan and Joel
Spruck, The existence of hypersurfaces of constant Gauss curvature
with prescribed boundary, J. Differential Geom. 62
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Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
(2007a:53001)
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Harold
Rosenberg, Hypersurfaces of constant curvature in space forms,
Bull. Sci. Math. 117 (1993), no. 2, 211–239. MR 1216008
(94b:53097)
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Smith G., Special Lagrangian curvature, arXiv:math/0506230.
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Smith G., Moduli of Flat Conformal Structures of Hyperbolic Type, arXiv:0804.0744.
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Smith G., The Non-Linear Dirichlet Problem in Hadamard Manifolds, arXiv:0908.3590
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Smith G., Constant Gaussian Curvature Hypersurfaces in Hadamard Manifolds, arXiv:
0912.0248
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Smith G., The Plateau Problem in Hadamard Manifolds, arXiv:1002.2982
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Neil
S. Trudinger and Xu-Jia
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(2004b:58017), http://dx.doi.org/10.1515/crll.2002.078
- 1.
- Andersson L., Barbot T., Béguin F., Zeghib A., Cosmological time versus CMC time in spacetimes of constant curvature, arXiv:math/0701452
- 2.
- Alexander S., Ghomi M., Wong J., Topology of Riemannian submanifolds with prescribed boundary, Duke Math. J. 152 (2010), no. 3, 533-565. MR 2654222 (2011d:53059)
- 3.
- Guan B., Spruck J., The existence of hypersurfaces of constant Gauss curvature with prescribed boundary, J. Differential Geom. 62 (2002), no. 2, 259-287. MR 1988505 (2004e:58019)
- 4.
- Labourie F., Un lemme de Morse pour les surfaces convexes (French), Invent. Math. 141 (2000), no. 2, 239-297. MR 1775215 (2001h:37059)
- 5.
- Petersen P., Riemannian geometry, Graduate Texts in Mathematics, 171, Springer, New York, 2006. MR 2243772 (2007a:53001)
- 6.
- Rosenberg H., Hypersurfaces of constant curvature in space forms, Bull. Sci. Math. 117 (1993), no. 2, 211-239. MR 1216008 (94b:53097)
- 7.
- Smith G., Special Lagrangian curvature, arXiv:math/0506230.
- 8.
- Smith G., Moduli of Flat Conformal Structures of Hyperbolic Type, arXiv:0804.0744.
- 9.
- Smith G., The Non-Linear Dirichlet Problem in Hadamard Manifolds, arXiv:0908.3590
- 10.
- Smith G., Constant Gaussian Curvature Hypersurfaces in Hadamard Manifolds, arXiv:
0912.0248
- 11.
- Smith G., The Plateau Problem in Hadamard Manifolds, arXiv:1002.2982
- 12.
- Trudinger N. S., Wang X., On locally convex hypersurfaces with boundary, J. Reine Angew. Math. 551 (2002), 11-32. MR 1932171 (2004b:58017)
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Additional Information
Graham Smith
Affiliation:
Instituto Nacional de Matemática Pura e Aplicada, Estrada Dona Castorina 110, Rio de Janeiro, Brazil 22460-320
DOI:
http://dx.doi.org/10.1090/S0002-9947-2012-05647-9
PII:
S 0002-9947(2012)05647-9
Keywords:
Dirichlet problem,
special Lagrangian curvature,
non-linear elliptic PDEs
Received by editor(s):
August 3, 2010
Posted:
October 31, 2012
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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