Mersenne banner

Livres, Actes et Séminaires du Centre Mersenne

  • Livres
  • Séminaires
  • Congrès
  • Tout
  • Auteur
  • Titre
  • Bibliographie
  • Plein texte
NOT
Entre et
  • Tout
  • Auteur
  • Titre
  • Date
  • Bibliographie
  • Mots-clés
  • Plein texte
  • Précédent
  • Séminaire de théorie spectrale et géométrie
  • Tome 26 (2007-2008)
  • p. 77-90
  • Suivant
Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature
Philippe G. LeFloch1
1 Université Pierre et Marie Curie (Paris VI) Laboratoire Jacques-Louis Lions (UMR CNRS 7598) 4 place Jussieu 75252 Paris (France)
Séminaire de théorie spectrale et géométrie, Tome 26 (2007-2008), pp. 77-90.
  • Résumé

We review recent work on the local geometry and optimal regularity of Lorentzian manifolds with bounded curvature. Our main results provide an estimate of the injectivity radius of an observer, and a local canonical foliations by CMC (Constant Mean Curvature) hypersurfaces, together with spatially harmonic coordinates. In contrast with earlier results based on a global bound for derivatives of the curvature, our method requires only a sup-norm bound on the curvature near the given observer.

  • Détail
  • Export
  • Comment citer
MR   Zbl
DOI : 10.5802/tsg.261
Classification : 83C05, 53C50, 53C12
Keywords: Lorentzian geometry, injectivity radius, constant mean curvature foliation, harmonic coordinates
Affiliations des auteurs :
Philippe G. LeFloch 1

1 Université Pierre et Marie Curie (Paris VI) Laboratoire Jacques-Louis Lions (UMR CNRS 7598) 4 place Jussieu 75252 Paris (France)
  • BibTeX
  • RIS
  • EndNote
@article{TSG_2007-2008__26__77_0,
     author = {Philippe G. LeFloch},
     title = {Injectivity radius and optimal regularity of {Lorentzian} manifolds with bounded curvature},
     journal = {S\'eminaire de th\'eorie spectrale et g\'eom\'etrie},
     pages = {77--90},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {26},
     year = {2007-2008},
     doi = {10.5802/tsg.261},
     mrnumber = {2654598},
     zbl = {1191.53052},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/tsg.261/}
}
TY  - JOUR
AU  - Philippe G. LeFloch
TI  - Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature
JO  - Séminaire de théorie spectrale et géométrie
PY  - 2007-2008
SP  - 77
EP  - 90
VL  - 26
PB  - Institut Fourier
PP  - Grenoble
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/tsg.261/
DO  - 10.5802/tsg.261
LA  - en
ID  - TSG_2007-2008__26__77_0
ER  - 
%0 Journal Article
%A Philippe G. LeFloch
%T Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature
%J Séminaire de théorie spectrale et géométrie
%D 2007-2008
%P 77-90
%V 26
%I Institut Fourier
%C Grenoble
%U https://proceedings.centre-mersenne.org/articles/10.5802/tsg.261/
%R 10.5802/tsg.261
%G en
%F TSG_2007-2008__26__77_0
Philippe G. LeFloch. Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature. Séminaire de théorie spectrale et géométrie, Tome 26 (2007-2008), pp. 77-90. doi : 10.5802/tsg.261. https://proceedings.centre-mersenne.org/articles/10.5802/tsg.261/
  • Bibliographie
  • Cité par

[1] M.T. Anderson, Convergence and rigidity of metrics under Ricci curvature bounds, Invent. Math. 102 (1990), 429–445. | MR | Zbl

[2] M.T. Anderson, Regularity for Lorentz metrics under curvature bounds, Jour. Math. Phys. 44 (2003), 2994–3012. | MR | Zbl

[3] L. Andersson and V. Moncrief, Elliptic-hyperbolic systems and the Einstein equations, Ann. Inst. Henri Poincaré 4 (2003), 1–34. | MR | Zbl

[4] L. Andersson and V. Moncrief, Future complete vacuum spacetimes, in “The Einstein equations and the large scale behavior of gravitational fields”, Birkhäuser, Basel, 2004, pp. 299–330. | MR | Zbl

[5] R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Commun. Math. Phys. 87 (1982), 131–152. | MR | Zbl

[6] A. Besse, Einstein manifolds, Ergebenisse Math. Series 3, Springer Verlag, 1987. | MR | Zbl

[7] J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Diff. Geom. 17 (1982), 15–53. | MR | Zbl

[8] B.-L. Chen and P.G. LeFloch, Injectivity radius estimates for Lorentzian manifolds, Commun. Math. Phys. 278 (2008), 679–713. | MR | Zbl

[9] B.-L. Chen and P.G. LeFloch, Local foliations and optimal regularity of Einstein spacetimes, submitted.

[10] D.M. DeTurck and J.L. Kazdan, Some regularity theorems in Riemannian geometry. Ann. Sci. École Norm. Sup. 14 (1981), 249–260. | Numdam | MR | Zbl

[11] C. Gerhardt, H-surfaces in Lorentzian manifolds, Commun. Math. Phys. 89 (1983), 523–533. | MR | Zbl

[12] S. Hawking and G.F. Ellis, The large scale structure of space-time, Cambridge Univ. Press, 1973. | MR | Zbl

[13] J. Jost and H. Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Manuscripta Math. 40 (1982), 27–77. | MR | Zbl

[14] S. Klainerman and I. Rodnianski, On the radius of injectivity of null hypersurfaces, J. Amer. Math. Soc. 21 (2008), 775–795. | MR

[15] S. Klainerman and I. Rodnianski, On the breakdown criterion in general relativity, preprint, 2008.

[16] R. Penrose, Techniques of differential topology in relativity, CBMS-NSF Region. Conf. Series Appli. Math., Vol. 7, 1972. | MR | Zbl

[17] P. Petersen, Convergence theorems in Riemannian geometry, in “Comparison Geometry” (Berkeley, CA, 1992–93), MSRI Publ. 30, Cambridge Univ. Press, 1997, pp. 167–202. | MR | Zbl

Cité par Sources :

Diffusé par : Publié par : Développé par :
  • Nous suivre