Convex spacelike hypersurfaces of constant curvature in de Sitter space

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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

Abstract
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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 and 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.
[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.
[3]	C. Gerhardt, "Curvature Problems,'' Series in Geometry and Topology, 39, International Press, Somerville, Mass., 2006.
[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.
[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.
[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.
[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
[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.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[3]	C. Gerhardt, "Curvature Problems,'' Series in Geometry and Topology, 39, International Press, Somerville, Mass., 2006.
[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.
[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.
[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.
[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
[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.
[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.
[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.
[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.
[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.
[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.

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