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September  2012, 17(6): 2225-2242. doi: 10.3934/dcdsb.2012.17.2225

Convex spacelike hypersurfaces of constant curvature in de Sitter space

Joel Spruck 1, and  Ling Xiao 1,

1. 

Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, United States

Received  October 2011 Revised  November 2011 Published  May 2012

We show that for a very general and natural class of curvature functions (for example the curvature quotients $(\sigma_n/\sigma_l)^{\frac{1}{n-l}}$) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space satisfying $f(\kappa)=\sigma \in (1,\infty)$ with a prescribed compact future asymptotic boundary $\Gamma$ at infinity has at least one smooth solution (if $l=1$ or $l=2$ there is uniqueness). This is the exact analogue of the asymptotic plateau problem in Hyperbolic space and is in fact a precise dual problem. By using this duality we obtain for free the existence of strictly convex solutions to the asymptotic Plateau problem for $\sigma_l=\sigma,\,1 \leq l < n$ in both de Sitter and Hyperbolic space.
Keywords: de Sitter, Hyperbolic space, fully nonlinear., constant curvature, asymptotic Plateau.
Mathematics Subject Classification: Primary: 53C21; Secondary: 35J6.
Citation: Joel Spruck, Ling Xiao. Convex spacelike hypersurfaces of constant curvature in de Sitter space. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2225-2242. doi: 10.3934/dcdsb.2012.17.2225
References:
[1]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equations, Comm. Pure Applied Math., 37 (1984), 369-402. doi: 10.1002/cpa.3160370306.  Google Scholar

[2]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of eigenvalues of the Hessians, Acta Math., 155 (1985), 261-301. doi: 10.1007/BF02392544.  Google Scholar

[3]

C. Gerhardt, "Curvature Problems,'' Series in Geometry and Topology, 39, International Press, Somerville, Mass., 2006.  Google Scholar

[4]

B. Guan and J. Spruck, Hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, Amer. J. Math., 122 (2000), 1039-1060. doi: 10.1353/ajm.2000.0038.  Google Scholar

[5]

B. Guan, J. Spruck and M. Szapiel, Hypersurfaces of constant curvature in hyperbolic space. I, J. Geom. Anal., 19 (2009), 772-795. doi: 10.1007/s12220-009-9086-7.  Google Scholar

[6]

B. Guan and J. Spruck, Hypersurfaces of constant curvature in hyperbolic space. II, J. European Math. Soc., 12 (2010), 797-817. doi: 10.4171/JEMS/215.  Google Scholar

[7]

B. Guan and J. Spruck, Convex hypersurfaces of constant curvature in Hyperbolic space, in "Surveys in Geometric Analysis and Relativity" (in honor of Richard Schoen), ALM 20, Higher Education Press, (2011), 241-257. Available from: arXiv:1010.4008 Google Scholar

[8]

S. W. Hawking and G. F. Ellis, "The Large Scale Structure of Spacetime," Cambridge Monographs on Mathematical Physics, No. 1, Cambridge Univ. Press, London-New York, 1973.  Google Scholar

[9]

S. Montiel, Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces, J. Math. Soc. Japan, 55 (2003), 915-938. doi: 10.2969/jmsj/1191418756.  Google Scholar

[10]

B. Nelli and J. Spruck, On existence and uniqueness of constant mean curvature hypersurfaces in hyperbolic space, in "Geometric Analysis and the Calculus of Variations,'' International Press, Cambridge, MA, (1996), 253-266.  Google Scholar

[11]

V. Oliker, A priori estimates of the principal curvatures of spacelike hypersurfaces in de Sitter space with applications to hypersurfaces in hyperbolic space, Amer. J. Math., 114 (1992), 605-626. doi: 10.2307/2374771.  Google Scholar

[12]

H. Rosenberg and J. Spruck, On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space, J. Differential Geom., 40 (1994), 379-409.  Google Scholar

[13]

J.-M. Schlenker, Hypersurfaces in $H^n$ and the space of its horospheres, Geom. Funct. Anal., 12 (2002), 395-435. doi: 10.1007/s00039-002-8252-x.  Google Scholar

show all references

References:
[1]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. I. Monge-Ampère equations, Comm. Pure Applied Math., 37 (1984), 369-402. doi: 10.1002/cpa.3160370306.  Google Scholar

[2]

L. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. III. Functions of eigenvalues of the Hessians, Acta Math., 155 (1985), 261-301. doi: 10.1007/BF02392544.  Google Scholar

[3]

C. Gerhardt, "Curvature Problems,'' Series in Geometry and Topology, 39, International Press, Somerville, Mass., 2006.  Google Scholar

[4]

B. Guan and J. Spruck, Hypersurfaces of constant mean curvature in hyperbolic space with prescribed asymptotic boundary at infinity, Amer. J. Math., 122 (2000), 1039-1060. doi: 10.1353/ajm.2000.0038.  Google Scholar

[5]

B. Guan, J. Spruck and M. Szapiel, Hypersurfaces of constant curvature in hyperbolic space. I, J. Geom. Anal., 19 (2009), 772-795. doi: 10.1007/s12220-009-9086-7.  Google Scholar

[6]

B. Guan and J. Spruck, Hypersurfaces of constant curvature in hyperbolic space. II, J. European Math. Soc., 12 (2010), 797-817. doi: 10.4171/JEMS/215.  Google Scholar

[7]

B. Guan and J. Spruck, Convex hypersurfaces of constant curvature in Hyperbolic space, in "Surveys in Geometric Analysis and Relativity" (in honor of Richard Schoen), ALM 20, Higher Education Press, (2011), 241-257. Available from: arXiv:1010.4008 Google Scholar

[8]

S. W. Hawking and G. F. Ellis, "The Large Scale Structure of Spacetime," Cambridge Monographs on Mathematical Physics, No. 1, Cambridge Univ. Press, London-New York, 1973.  Google Scholar

[9]

S. Montiel, Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces, J. Math. Soc. Japan, 55 (2003), 915-938. doi: 10.2969/jmsj/1191418756.  Google Scholar

[10]

B. Nelli and J. Spruck, On existence and uniqueness of constant mean curvature hypersurfaces in hyperbolic space, in "Geometric Analysis and the Calculus of Variations,'' International Press, Cambridge, MA, (1996), 253-266.  Google Scholar

[11]

V. Oliker, A priori estimates of the principal curvatures of spacelike hypersurfaces in de Sitter space with applications to hypersurfaces in hyperbolic space, Amer. J. Math., 114 (1992), 605-626. doi: 10.2307/2374771.  Google Scholar

[12]

H. Rosenberg and J. Spruck, On the existence of convex hypersurfaces of constant Gauss curvature in hyperbolic space, J. Differential Geom., 40 (1994), 379-409.  Google Scholar

[13]

J.-M. Schlenker, Hypersurfaces in $H^n$ and the space of its horospheres, Geom. Funct. Anal., 12 (2002), 395-435. doi: 10.1007/s00039-002-8252-x.  Google Scholar

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