Advertisement

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beem J.K., Ehrlich P.E., Easley K.L.: Global Lorentzian geometry, 2nd edn. Marcel Dekker Inc., New York (1996)MATHGoogle Scholar
  2. 2.
    Bernal A.N., Sánchez M.: Globally hyperbolic spacetimes can be defined as “causal” instead of “strongly causal”. Classical Quantum Gravity. 24, 745–749 (2007)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Cheeger J., Gromov M., Taylor M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom. 17, 15–53 (1982)MathSciNetMATHGoogle Scholar
  4. 4.
    Chruściel, P.T. : Elements of causality theory. Preprint arxiv:1110.6706Google Scholar
  5. 5.
    Ehrlich P.E.: Comparison theory in Lorentzian geometry, Lecture notes. Isaac Newton Institute, Cambridge (2005)Google Scholar
  6. 6.
    Ehrlich P.E., Jung Y.-T., Kim S.-B.: Volume comparison theorems for Lorentzian manifolds. Geom. Dedicata 73, 39–56 (1998)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Ehrlich P.E., Sánchez M.: Some semi-Riemannian volume comparison theorems. Tohoku Math. J. 2(52), 331–348 (2000)CrossRefGoogle Scholar
  8. 8.
    Eschenburg J.H., Heintze E.: Comparison theory for Riccati equations. Manuscripta Math. 68, 209–214 (1990)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Friedlander F.G.: The wave equation on a curved space-time. Cambridge University Press, Cambridge (1975)MATHGoogle Scholar
  10. 10.
    Galloway G.J.: Curvature, causality and completeness in space-times with causally complete spacelike slices. Math. Proc. Cambridge Philos. Soc. 99, 367–375 (1986)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Grant J. D. E.: A synthetic singularity theorem. (2012) (in preparation)Google Scholar
  12. 12.
    Hawking S.W., Ellis G.F.R.: The large scale structure of space-time. Cambridge University Press, London (1973)MATHCrossRefGoogle Scholar
  13. 13.
    Heintze E., Karcher H.: A general comparison theorem with applications to volume estimates for submanifolds. Ann. Sci. École Norm. Sup. 4(11), 451–470 (1978)MathSciNetGoogle Scholar
  14. 14.
    Lott J., Villani C.: Ricci curvature for metric-measure spaces via optimal transport. Ann. of Math. 169, 903–991 (2009)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Minguzzi, E., Sánchez M.: The causal hierarchy of spacetimes. In: Alekseevsky, D., Baum, H. (eds.) Recent developments in pseudo-Riemannian geometry. EMS, Zürich (2008)Google Scholar
  16. 16.
    Ohta S.I.: On the measure contraction property of metric measure spaces. Comment. Math. Helv. 82, 805–828 (2007)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    O’Neill B.: Semi-Riemannian geometry. Academic Press Inc., New York (1983)MATHGoogle Scholar
  18. 18.
    Sakai T.: Riemannian geometry. American Mathematical Society, Providence (1996)MATHGoogle Scholar
  19. 19.
    Sturm K.-T.: On the geometry of metric measure spaces. II.. 196, 133–177 (2006)MathSciNetMATHGoogle Scholar
  20. 20.
    Treude, J.-H.: Ricci curvature comparison in Riemannian and Lorentzian geometry. Diploma thesis. Universität Freiburg. Freiburg. http://www.freidok.uni-freiburg.de/volltexte/8405. 2011
  21. 21.
    Warner F.W.: Extensions of the Rauch comparison theorem to submanifolds. Trans. Amer. Math. Soc. 122, 341–356 (1966)MathSciNetMATHGoogle Scholar
  22. 22.
    Zhu, S.: The comparison geometry of Ricci curvature. In: Comparison geometry (Berkeley, CA, 1993–1994), pp. 221–262. Cambridge University Press, Cambridge (1997)Google Scholar

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

Personalised recommendations