Stable maximal hypersurfaces in Lorentzian spacetimes

Abstract

We study the geometry of stable maximal hypersurfaces in a variety of spacetimes satisfying various physically relevant curvature assumptions, for instance the Timelike Convergence Condition (TCC). We characterize stability when the target space has constant sectional curvature as well as give sufficient conditions on the geometry of the ambient spacetime (e.g., the validity of TCC) to ensure stability. Some rigidity results and height estimates are also proven in GRW spacetimes. In the last part of the paper we consider $k$-stability of spacelike hypersurfaces, a concept related to mean curvatures of higher orders.

arXiv comment30 pages. This is a pre-print of an article published in Nonlinear Analysis. The final authenticated version is available online at: https://doi.org/10.1016/j.na.2018.09.009

References in corpus (6)

• Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes
• Calabi-Bernstein results for maximal surfaces in Lorentzian product spaces
• From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity
• On maximal hypersurfaces in Lorentz manifolds admitting a parallel lightlike vector field
• On the $r-$stability of spacelike hypersurfaces
• Uniqueness of complete maximal hypersurfaces in spatially open $(n+1)$-dimensional Robertson-Walker spacetimes with flat fiber