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Spacelike hypersurfaces in standard static spacetimes

  • Giulio ColomboEmail author
  • José A. S. Pelegrín
  • Marco Rigoli
Research Article

Abstract

In this work we study spacelike hypersurfaces immersed in spatially open standard static spacetimes with complete spacelike slices. Under appropriate lower bounds on the Ricci curvature of the spacetime in directions tangent to the slices, we prove that every complete CMC hypersurface having either bounded hyperbolic angle or bounded height is maximal. Our conclusions follow from general mean curvature estimates for spacelike hypersurfaces. In case where the spacetime is a Lorentzian product with spatial factor of nonnegative Ricci curvature and sectional curvatures bounded below, we also show that a complete maximal hypersurface not intersecting a spacelike slice is itself a slice. This result is obtained from a gradient estimate for parametric maximal hypersurfaces.

Keywords

Standard static spacetime Complete spacelike hypersurface Geometric estimates Calabi–Bernstein type result 

Mathematics Subject Classification

53C42 53C50 58J05 

Notes

Acknowledgements

The second author is partially supported by Spanish MINECO and ERDF project MTM2016-78807-C2-1-P.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di MatematicaUniversità degli Studi di MilanoMilanItaly
  2. 2.Departamento de Geometría y TopologíaUniversidad de GranadaGranadaSpain

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