Entire spacelike -graphs in Lorentzian product spaces

In this work we establish suffcient conditions to ensure that an entire spacelike graph immersed with constant mean curvature in a Lorentzian product space, whose Riemannian fiber has sectional curvature bounded from below, must be a trivial slice of the ambient space

Compact Spacelike Hypersurfaces with Constant Mean Curvature in the Antide Sitter Space

We obtain a height estimate concerning to a compact spacelike hypersurface Σ n immersed with constant mean curvature H in the anti-de Sitter space H n 1 1 , when its boundary ∂Σ is contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic space H n . Our estimate depends only on the value of H and on the geometry of ∂Σ. As applications of our estimate, we obtain a characterization of hyperbolic domains of H n 1 1 and nonexistence results in connection with such types of hypersurfaces

Compact Spacelike Hypersurfaces with Constant Mean Curvature in the Antide Sitter Space

We obtain a height estimate concerning to a compact spacelike hypersurface Σn immersed with constant mean curvature H in the anti-de Sitter space ℍ1n+1, when its boundary ∂Σ is contained into an umbilical spacelike hypersurface of this spacetime which is isometric to the hyperbolic space ℍn. Our estimate depends only on the value of H and on the geometry of ∂Σ. As applications of our estimate, we obtain a characterization of hyperbolic domains of ℍ1n+1 and nonexistence results in connection with such types of hypersurfaces

Local rigidity, bifurcation, and stability of Hf-hypersurfaces in weighted Killing warped products

In a weighted Killing warped product Mn f ×ρR with warping metric h , iM+ρ2 dt, where the warping function ρ is a real positive function defined on Mn and the weighted function f does not depend on the parameter t ∈ R, we use equivariant bifurcation theory in order to establish sufficient conditions that allow us to guarantee the existence of bifurcation instants, or the local rigidity for a family of open sets {Ωγ}γ∈I whose boundaries ∂Ωγ are hypersurfaces with constant weighted mean curvature. For this, we analyze the number of negative eigenvalues of a certain Schrödinger operator and study its evolution. Furthermore, we obtain a characterization of a stable closed hypersurface x: Σn ,→ Mn f ×ρ R with constant weighted mean curvature in terms of the first eigenvalue of the f-Laplacian of Σn