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Constant mean curvature spacelike hypersurfaces in standard static spaces: rigidity and parabolicity. (English) Zbl 07276077
Summary: Our purpose in this paper is investigate the geometry of complete constant mean curvature spacelike hypersurfaces immersed in a standard static space, that is, a Lorentzian manifold endowed with a globally defined timelike Killing vector field. In this setting, supposing that the ambient space is a warped product of the type $$M^n\times_{\rho}\mathbb{R}_1$$ whose Riemannian base $$M^n$$ has nonnegative sectional curvature and the warping function $$\rho$$ is convex on $$M^n$$, we use the generalized maximum principle of Omori-Yau in order to establish rigidity results concerning these spacelike hypersurfaces. We also study the parabolicity of maximal spacelike surfaces in $$M^2\times_{\rho}\mathbb{R}_1$$ and we obtain uniqueness results for entire Killing graphs constructed over $$M^n$$.
##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53B30 Local differential geometry of Lorentz metrics, indefinite metrics 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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