Skip to main content
SpringerLink
Log in
Book cover

The Einstein Equations and the Large Scale Behavior of Gravitational Fields pp 347–377Cite as

Birkhäuser

Cheeger-Gromov Theory and Applications to General Relativity

  • Conference paper

Abstract

This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a given manifold M, and some recent applications of this theory to general relativity. The basic point of view of convergence/degeneration described here originates in the work of Gromov, cf. [31]—[33], with important prior work of Cheeger [16], leading to the joint work of [18].

This Cheeger-Gromov theory assumes L ∞ bounds on the full curvature tensor. For reasons discussed below, we focus mainly on the generalizations of this theory to spaces with L ∞, (or L p) bounds on the Ricci curvature. Although versions of the results described hold in any dimension, for the most part we restrict the discussion to 3 and 4 dimensions, where stronger results hold and the applications to general relativity are most direct. The first three sections survey the theory in Riemannian geometry, while the last three sections discuss applications to general relativity.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. M. Anderson, Convergence and rigidity of manifolds under Ricci curvature bounds, Inventiones Math., 106,(1990) 429–445.

    Article  ADS  Google Scholar 

  2. M. Anderson, Short geodesics and gravitational instantons, Jour. Diff. Geom., 31, (1990), 265–275.

    MATH  Google Scholar 

  3. M. Anderson and J. Cheeger, C α compactness for manifolds with Ricci curvature and injectivity radius bounded below, Jour. Diff. Geom., 35, (1992), 265–281.

    MathSciNet  MATH  Google Scholar 

  4. M. Anderson, Hausdorff perturbations of Ricci-flat metrics and the splitting theorem, Duke Math. Jour., 68, (1992), 67–82.

    Article  MATH  Google Scholar 

  5. M. Anderson, Extrema of curvature functionals on the space of metrics on 3-manifolds, Calc. Var. & P.D.E., 5, (1997), 199–269.

    Article  MATH  Google Scholar 

  6. M. Anderson, On stationary vacuum solutions to the Einstein equations, Ann. Henri Poincaré, 1, (2000), 977–994.

    Article  ADS  MATH  Google Scholar 

  7. M. Anderson, On the structure of solutions to the static vacuum Einstein equations, Ann. Henri Poincaré, 1, (2000), 995–1042.

    Article  ADS  MATH  Google Scholar 

  8. M. Anderson, On long-time evolution in general relativity and geometrization of 3-manifolds, Comm. Math. Phys., 222, (2001), 533–567.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  9. M. Anderson, Regularity for Lorentz metrics under curvature bounds, Jour. Math. Physics, 44, (2003), 2994–3012.

    Article  ADS  MATH  Google Scholar 

  10. L. Andersson and V. Moncrief, Future complete vacuum spacetimes, this volume and gr-qc/0303045.

    Google Scholar 

  11. R. Bartnik, Regularity of variational maximal surfaces, Acta Mathematica, 161, (1988), 145–181.

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Bartnik, Remarks on cosmological space-times and constant mean curvature surfaces, Comm Math. Phys., 117, (1988), 615–624.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  13. J. Beem, P. Ehrlich and K. Easley, Global Lorentzian Geometry, 2nd Edition, Marcel Dekker, New York, (1996).

    MATH  Google Scholar 

  14. R. Beig and W. Simon, Proof of a multipole conjecture due to Geroch, Comm. Math. Phys., 78, (1980), 75–82.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. J. Bičák, Selected solutions of Einstein’s field equations: their role in General Relativity and Astrophysics, in Einstein’s Field Equations and their Physical Applications, Lecture Notes in Physics, 540, Springer Verlag, Berlin, (2000), 1–126.

    Google Scholar 

  16. J. Cheeger, Finiteness theorems for Riemannian manifolds, Amer. Jour. Math., 92, (1970), 61–75.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Cheeger and D. Gromoll, The splitting theorem for manifolds of non-negative Ricci curvature, Jour. Diff. Geom., 6, (1971), 119–128.

    MathSciNet  MATH  Google Scholar 

  18. J. Cheeger and M. Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded, I, II, Jour. Diff. Geom., 23, (1986), 309–346 and 32, (1990), 269–298.

    Google Scholar 

  19. Y. Choquet-Bruhat, Solutions C∞ déequations hyperboliques non-linéaires, C.R. Acad. Sci. Paris, Sér. A-B, 272, (1971), A386–A388.

    MathSciNet  Google Scholar 

  20. Y. Choquet-Bruhat and V. Moncrief, Future global in time Einsteinian spacetimes with U(1) isometry group, Ann. Henri Poincaré, 2, (2001), 1007–1064.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. D. Christodoulou and S. Klainerman, The Global Nonlinear Stability of the Minkowski Space, Princeton Univ. Press, Princeton, (1993).

    MATH  Google Scholar 

  22. C.J.S. Clarke, The Analysis of Space-Time Singularities, Camb. Lect. Notes in Physics, 1, Camb. Univ. Press, (1993).

    Google Scholar 

  23. D. Deturck and J. Kazdan, Some regularity theorems in Riemannian geometry, Ann. Sci. Ecole Norm. Sup, 14, (1981), 249–260.

    MathSciNet  MATH  Google Scholar 

  24. A. Einstein and W. Pauli, Non-existence of regular stationary solutions of relativistic field equations, Annals of Math., ser. 2, 44, (1943), 131–137.

    Article  MathSciNet  MATH  Google Scholar 

  25. A. Fischer and V. Moncrief, The reduced Hamiltonian of general relativity and the a-constant of conformal geometry, in Math. and Quantum Aspects of Relativity and Cosmology, S. Cotsakis and G.W. Gibbons, (Eds.), Springer Lecture Notes in Physics, 537, (2000), 70–101.

    Chapter  Google Scholar 

  26. H. Friedrich, On the existence of n-geodesically complete or future complete solutions of Einstein’s field equations with smooth asymptotic structure, Comm Math. Phys., 107, (1986), 587–609.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. K. Fukaya, Collapsing Riemannian manifolds to ones of lower dimension, Jour. Diff. Geom., 25, (1987), 139–156.

    MathSciNet  MATH  Google Scholar 

  28. D. Garfinkle and M. Melvin, Generalized magnetic universe solutions, Phys. Rev. D50, (1994), 3859–3866.

    MathSciNet  ADS  Google Scholar 

  29. C. Gerhardt, H-surfaces in Lorentzian manifolds, Comm Math. Phys., 89, (1983), 523–553.

    Article  MathSciNet  ADS  MATH  Google Scholar 

  30. G.W. Gibbons and S.W. Hawking, Classification of gravitational instanton symmetries, Comm. Math. Phys., 66, (1979), 291–310.

    Article  MathSciNet  ADS  Google Scholar 

  31. M. Gromov, Structures Métriques Pour Les Variétés Riemanniennes, Cédic-Nathan, Paris, (1981).

    MATH  Google Scholar 

  32. M. Gromov, Almost flat manifolds, Jour. Diff. Geom., 13, (1978), 231–241.

    MathSciNet  MATH  Google Scholar 

  33. M. Gromov, Volume and bounded cohomology, Publ. Math. I.H.E.S., 56, (1982), 5–99.

    MathSciNet  MATH  Google Scholar 

  34. C. Gundlach, Critical phenomena in gravitational collapse, Living Reviews in Relativity, 2:4, (1999), http://www.livingreviews.org

    MathSciNet  ADS  Google Scholar 

  35. S. Hawking and G. Ellis, The Large Scale Structure of Space-Time, Cambridge Univ. Press, London, (1973).

    Book  Google Scholar 

  36. J. Jost and H. Karcher, Geometrische Methoden zur Gewinnung von a priori Schranken für harmonische Abbildungen, Manuscripta Math., 40, (1982), 21–71.

    Article  MathSciNet  Google Scholar 

  37. A. Lichnerowicz, Théories Relativistes de la Gravitation et de l’Electromagnétisme, Masson, Paris, (1955).

    MATH  Google Scholar 

  38. J. Marsden and F. Tipler, Maximal hypersurfaces and foliations of constant mean curvature in general relativity, Physics Reports, 66:3, (1980), 109–139.

    Article  MathSciNet  Google Scholar 

  39. M. Melvin, Pure magnetic and electric geons, Physics Letters, 8:1, (1964), 65–68.

    Article  MathSciNet  Google Scholar 

  40. R. Myers, Higher-dimensional black holes in compactified space-times, Phys. Rev. D., 35, (1987), 455–466.

    Article  MathSciNet  ADS  Google Scholar 

  41. P. Orlik, Seifert Manifolds, Springer Lecture Notes in Math., 291, Springer Verlag, New York, (1972).

    Google Scholar 

  42. R. Penrose, Any space-time has a plane-wave as a limit, in Differential Geometry and Relativity, Ed. Cahan and Flato, (1976), 271–275.

    Chapter  Google Scholar 

  43. P. Petersen, Riemannian Geometry, Graduate Texts in Mathematics, Springer Verlag, New York, (1998).

    MATH  Google Scholar 

  44. A. Rendall, Constant mean curvature foliations in cosmological space-times, Hely. Physica Acta, 69:4, (1996), 490–500.

    MathSciNet  ADS  MATH  Google Scholar 

  45. A. Rendall, Compact null hypersurfaces and collapsing Riemannian manifolds, Math. Nachrichten, 193, (1998), 111–118.

    Article  MathSciNet  MATH  Google Scholar 

  46. W. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc., 6, (1982), 357–381.

    Article  MathSciNet  MATH  Google Scholar 

  47. F. Tipler, C. Clarke and G. Ellis, Singularities and horizons: a review article, in Gen. Rel. and Gravitation, vol. 2, A. Held, (Ed.), Plenum Press, New York, (1980), 87–206.

    Google Scholar 

  48. F. Waldhausen, Eine Klasse von 3-dimensionalen Mannifaltigkeiten, I, II, Invent. Math. 3, (1967), 308–333 and 4, (1967), 87–117.

    Article  MathSciNet  ADS  Google Scholar 

  49. G. Perelman, The entropy formula for the Ricci flow and its geometric applications, math.DG/0211159.

    Google Scholar 

  50. G. Perelman, Ricci flow with surgery on three-manifolds, math.DG/0303109.

    Google Scholar 

  51. G. Perelman, Finite extinction time for the solutions to the Ricci flow on certain three-manifolds, math.DG/0307245.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, S.U.N.Y. at Stony Brook, Stony Brook, N.Y., 11794-3651, USA

    Michael T. Anderson

Authors
  1. Michael T. Anderson
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Laboratoire de Mathématiques et de Physique Théorique, CNRS UMR 6083, Avenue Grammont, 37200, Tours, France

    Piotr T. Chruściel

  2. Max-Planck-Institut für Gravitationsphysik, Am Mühlenberg 1, 14467, Golm, Germany

    Helmut Friedrich

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Basel AG

About this paper

Cite this paper

Anderson, M.T. (2004). Cheeger-Gromov Theory and Applications to General Relativity. In: Chruściel, P.T., Friedrich, H. (eds) The Einstein Equations and the Large Scale Behavior of Gravitational Fields. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7953-8_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-7953-8_10

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9634-4

  • Online ISBN: 978-3-0348-7953-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation