Convex Spacelike Hypersurfaces of Constant Curvature in de Sitter Space

Details Convex Spacelike Hypersurfaces of Constant Curvature in de Sitter Space Convex Spacelike Hypersurfaces of Constant Curvature in de Sitter Space

2012-03-26

arxiv.org/abs/1203.5739v1

Mathematics

Differential Geometry

24 pages

Scientific paper

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 deSitter and Hyperbolic space.

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