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Annals of Global Analysis and Geometry

, Volume 43, Issue 3, pp 233–251 | Cite as

Volume comparison for hypersurfaces in Lorentzian manifolds and singularity theorems

  • Jan-Hendrik Treude
  • James D. E. Grant
Article

Abstract

We develop area and volume comparison theorems for the evolution of spacelike, acausal, causally complete hypersurfaces in Lorentzian manifolds, where one has a lower bound on the Ricci tensor along timelike curves, and an upper bound on the mean curvature of the hypersurface. Using these results, we give a new proof of Hawking’s singularity theorem.

Keywords

Lorentzian manifolds Comparison geometry Singularity theorems 

Mathematics Subject Classification (2000)

53C50 83C75 53C80 51P05 

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität RegensburgRegensburgGermany
  2. 2.Gravitationsphysik, Fakultät für PhysikUniversität WienWienAustria

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