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On the radius of injectivity of null hypersurfaces


Authors: Sergiu Klainerman and Igor Rodnianski
Journal: J. Amer. Math. Soc. 21 (2008), 775-795
MSC (2000): Primary 35J10
DOI: https://doi.org/10.1090/S0894-0347-08-00592-4
Published electronically: March 18, 2008
MathSciNet review: 2393426
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Abstract: We investigate the regularity of past (future) boundaries of points in regular, Einstein vacuum spacetimes. We provide conditions, expressed in terms of a space-like foliation and which imply, in particular, uniform $ L^2$ bounds for the curvature tensor, sufficient to ensure the local nondegeneracy of these boundaries. More precisely we provide a uniform lower bound on the radius of injectivity of the null boundaries $ \mathcal{N}^{\pm}(p)$ of the causal past (future) sets $ \mathcal{J}^{\pm}(p)$. Such lower bounds are essential in understanding the causal structure and the related propagation properties of solutions to the Einstein equations. They are particularly important in construction of an effective Kirchoff-Sobolev type parametrix for solutions of wave equations on $ \mathbf{M}$. Such parametrices are used by the authors in a forthcoming paper to prove a large data break-down criterion for solutions of the Einstein vacuum equations.


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Additional Information

Sergiu Klainerman
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: seri@math.princeton.edu

Igor Rodnianski
Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
Email: irod@math.princeton.edu

DOI: https://doi.org/10.1090/S0894-0347-08-00592-4
Received by editor(s): March 5, 2006
Published electronically: March 18, 2008
Additional Notes: The first author is supported by NSF grant DMS-0070696
The second author is partially supported by NSF grant DMS-0406627
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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