Complete Maximal Hypersurfaces Research Articles

Uniqueness of complete maximal hypersurfaces in spatially open $$(n+1)$$ ( n + 1 ) -dimensional Robertson–Walker spacetimes with flat fiber

In this paper, under natural geometric and physical assumptions we provide new uniqueness and non-existence results for complete maximal hypersurfaces in spatially open Robertson-Walker spacetimes whose fiber is flat. Moreover, our results are applied to relevant spacetimes as the steady state spacetime, Einstein-de Sitter spacetime and radiation models.

Complete maximal hypersurfaces in certain spatially open generalized Robertson–Walker spacetimes

Uniqueness and non-existence results on maximal hypersurfaces lying between two spacelike slices in open generalized Robertson–Walker spacetimes are given. They are obtained from a known generalized maximum principle applied to a distinguished smooth function on a complete maximal hypersurface.

A new approach for uniqueness of complete maximal hypersurfaces in spatially parabolic GRW spacetimes

Complete maximal hypersurfaces in Generalized Robertson–Walker spacetimes whose fiber has a parabolic universal Riemannian covering are studied. Under natural boundedness assumptions, a spacelike hypersurface is shown to be parabolic when it is endowed with a conformal metric. This fact allows us to get, for complete maximal hypersurfaces, several uniqueness results. As an application, all the entire solutions to the maximal hypersurface equation under certain constrains are obtained, solving new Calabi–Bernstein problems.

Uniqueness of complete maximal hypersurfaces in spatially parabolic generalized Robertson–Walker spacetimes

A new technique for the study of noncompact complete spacelike hypersurfaces in generalized Robertson–Walker (GRW) spacetimes whose fiber is a parabolic Riemannian manifold is introduced. This class of spacetimes allows us to model open universes which extend to spacelike closed GRW spacetimes from the viewpoint of the geometric analysis of the fiber, and which, unlike those spacetimes, could be compatible with the holographic principle. First, under reasonable assumptions on the restriction of the warping function to the spacelike hypersurface and on the hyperbolic angle between the unit normal vector field and a certain timelike vector field, a complete spacelike hypersurface in a spatially parabolic GRW spacetime is shown to be parabolic, and the existence of a simply connected parabolic spacelike hypersurface in a GRW spacetime also leads to the parabolicity of its fiber. Then, all the complete maximal hypersurfaces in spatially parabolic GRW spacetimes are determined in several cases, extending, in particular, to this family of open cosmological models several well-known uniqueness results for the case of spatially closed GRW spacetimes. Moreover, new Calabi–Bernstein problems are solved.