American Institute of Mathematical Sciences

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

Convex spacelike hypersurfaces of constant curvature in de Sitter space

 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.
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:
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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.  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.  doi: 10.1007/BF02392544.  Google Scholar [3] C. Gerhardt, "Curvature Problems,'', Series in Geometry and Topology, 39 (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.  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.  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.  doi: 10.4171/JEMS/215.  Google Scholar [7] B. Guan and J. Spruck, Convex hypersurfaces of constant curvature in Hyperbolic space,, in, (2011), 241.   Google Scholar [8] S. W. Hawking and G. F. Ellis, "The Large Scale Structure of Spacetime,", Cambridge Monographs on Mathematical Physics, (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.  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, (1996), 253.   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.  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.   Google Scholar [13] J.-M. Schlenker, Hypersurfaces in $H^n$ and the space of its horospheres,, Geom. Funct. Anal., 12 (2002), 395.  doi: 10.1007/s00039-002-8252-x.  Google Scholar
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