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Complete non-compact spacelike hypersurfaces of constant mean curvature in de Sitter spaces. (English) Zbl 1049.53044
The paper under review focuses on constructions of complete non-compact space-like hypersurfaces in de Sitter space \(\mathbb S^{n+1}_1\) with constant mean curvature \(H > 1\) satisfying some asymptotic future boundary condition by using the fact that such a hypersurface can be obtained as a limit of constant mean curvature graphs over compact domains contained in time slices of the steady state space \(\mathcal H^{n+1}\) of \(\mathbb S^{n+1}_1.\) In order ro make estimations for the height and the slope of the space-like graphs with constant curvature, the half space model is used for the open set of de Sitter space. By making use of the estimations obtained, some existence and uniqueness theorems for complete non-compact constant mean curvature space-like hypersurfaces in de Sitter spaces with some special asymptotic future boundary behavior are considered, i.e., the solution of the following Dirichlet problem is examined \[ {\text{ div}}\Bigl(\frac{\nabla f}{\sqrt{1-| \nabla f| ^2}}\Bigl)+ \frac{n}{f}\Bigl(H-\frac{1}{\sqrt{1-| \nabla f| ^2}} \Bigl)=0 \] where \(| \nabla f| ^2 < 1\) on \(\text{int}(\Omega)\), \(f=t\) on \(\partial \Omega\), \(H \geq 1,\Omega\) is a compact domain in \(\mathbb R^n\) with mean convex boundary and \(t \in (0,\infty)\) and some results are obtained about the existence of solutions to have space-like constant mean curvature graphs over compact domains in horizontal time slices with mean convex boundaries.

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
35Q75 PDEs in connection with relativity and gravitational theory
83C99 General relativity
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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