A New Technique for the Study of Complete Maximal Hypersurfaces in Certain Open Generalized Robertson–Walker Spacetimes
An (n + 1)-dimensional Generalized Robertson–Walker (GRW) spacetime such that the universal Riemannian covering of the fiber is parabolic (thus so is the fiber) is said to be spatially parabolic. This class of spacetimes allows to model open relativistic universes which extend to the spatially closed GRW spacetimes from the viewpoint of the geometric-analysis of the fiber and which are not incompatible with certain cosmological principle. We explain here a new technique for the study of non-compact complete spacelike hypersurfaces in such spacetimes. Thus, a complete spacelike hypersurface in a spatially parabolic GRW spacetime inherits the parabolicity, whenever some boundedness 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 are assumed. Conversely, the existence of a simply connected parabolic spacelike hypersurface, under the previous assumptions, in a GRW spacetime also leads to its spatial parabolicity. Then, all the complete maximal hypersurfaces in a spatially parabolic GRW spacetime are determined in several cases, extending known uniqueness results. Finally, all the entire solutions of the maximal hypersurface equation on a parabolic Riemannian manifold are found in several cases, solving new Calabi–Bernstein problems.
KeywordsRiemannian Manifold Spacelike Hypersurface Warping Function Normal Vector Field Hyperbolic Angle
Supported by the Spanish MICINN Grant with FEDER funds MTM2010-18099, the Junta de Andalucía Regional Grant with FEDER funds P09-FQM-4496, National Institute for Mathematical Sciences, Daejeon, Korea, and Grassmannian Research Group of the Dep. of Mathematics of the Kyungpook National University, Daegu, Korea. The author would like also to express his sincere thanks to Prof. Y.J. Suh and Dr. Hyunjin Lee.
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