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- Title
CONVEX SPACELIKE HYPERSURFACES OF CONSTANT CURVATURE IN DE SITTER SPACE.
- Authors
Spruck, Joel; Ling Xiao
- Abstract
We show that for a very general and natural class of curvature functions (for example the curvature quotients (σn/σl)1/n-1 ) the problem of finding a complete spacelike strictly convex hypersurface in de Sitter space satisfying f(k) = σ ∊ (1, ∞ ) with a prescribed compact future asymptotic boundary Γ 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 σl = σ, 1 ≤ l < n in both de Sitter and Hyperbolic space.
- Subjects
CONVEX surfaces; HYPERSURFACES; SPACES of constant curvature; HYPERBOLIC spaces; PLATEAU'S problem
- Publication
Discrete & Continuous Dynamical Systems - Series B, 2012, Vol 17, Issue 6, p2225
- ISSN
1531-3492
- Publication type
Article
- DOI
10.3934/dcdsb.2012.17.2225