## On the radius of injectivity of null hypersurfaces

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- by Sergiu Klainerman and Igor Rodnianski
- J. Amer. Math. Soc.
**21**(2008), 775-795 - DOI: https://doi.org/10.1090/S0894-0347-08-00592-4
- Published electronically: March 18, 2008
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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.## References

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## Bibliographic Information

**Sergiu Klainerman**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- MR Author ID: 102350
- Email: seri@math.princeton.edu
**Igor Rodnianski**- Affiliation: Department of Mathematics, Princeton University, Princeton, New Jersey 08544
- Email: irod@math.princeton.edu
- 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 - © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: J. Amer. Math. Soc.
**21**(2008), 775-795 - MSC (2000): Primary 35J10
- DOI: https://doi.org/10.1090/S0894-0347-08-00592-4
- MathSciNet review: 2393426