Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature

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)

Séminaire de théorie spectrale et géométrie (2007-2008)

  • Volume: 26, page 77-90
  • ISSN: 1624-5458

Abstract

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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.

How to cite

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LeFloch, Philippe G.. "Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature." Séminaire de théorie spectrale et géométrie 26 (2007-2008): 77-90. <http://eudml.org/doc/11240>.

@article{LeFloch2007-2008,
abstract = {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.},
affiliation = {Université Pierre et Marie Curie (Paris VI) Laboratoire Jacques-Louis Lions (UMR CNRS 7598) 4 place Jussieu 75252 Paris (France)},
author = {LeFloch, Philippe G.},
journal = {Séminaire de théorie spectrale et géométrie},
keywords = {Lorentzian geometry; injectivity radius; constant mean curvature foliation; harmonic coordinates},
language = {eng},
pages = {77-90},
publisher = {Institut Fourier},
title = {Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature},
url = {http://eudml.org/doc/11240},
volume = {26},
year = {2007-2008},
}

TY - JOUR
AU - LeFloch, Philippe G.
TI - Injectivity radius and optimal regularity of Lorentzian manifolds with bounded curvature
JO - Séminaire de théorie spectrale et géométrie
PY - 2007-2008
PB - Institut Fourier
VL - 26
SP - 77
EP - 90
AB - 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.
LA - eng
KW - Lorentzian geometry; injectivity radius; constant mean curvature foliation; harmonic coordinates
UR - http://eudml.org/doc/11240
ER -

References

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  12. S. Hawking and G.F. Ellis,The large scale structure of space-time, Cambridge Univ. Press, 1973. Zbl0265.53054MR424186
  13. J. Jost and H. Karcher, Geometrische Methoden zur Gewinnung von a-priori-Schranken für harmonische Abbildungen, Manuscripta Math. 40 (1982), 27–77. Zbl0502.53036MR679120
  14. S. Klainerman and I. Rodnianski, On the radius of injectivity of null hypersurfaces, J. Amer. Math. Soc. 21 (2008), 775–795. Zbl1198.53057MR2393426
  15. S. Klainerman and I. Rodnianski, On the breakdown criterion in general relativity, preprint, 2008. Zbl1203.35084
  16. R. Penrose,Techniques of differential topology in relativity, CBMS-NSF Region. Conf. Series Appli. Math., Vol. 7, 1972. Zbl0321.53001MR469146
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