## On the radius of injectivity of null hypersurfaces

HTML articles powered by AMS MathViewer

- by Sergiu Klainerman and Igor Rodnianski PDF
- J. Amer. Math. Soc.
**21**(2008), 775-795 Request permission

## 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

- Michael T. Anderson,
*Regularity for Lorentz metrics under curvature bounds*, J. Math. Phys.**44**(2003), no. 7, 2994–3012. MR**1982778**, DOI 10.1063/1.1580199 - Michael T. Anderson,
*Cheeger-Gromov theory and applications to general relativity*, The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, pp. 347–377. MR**2098921** - Michael T. Anderson,
*On long-time evolution in general relativity and geometrization of 3-manifolds*, Comm. Math. Phys.**222**(2001), no. 3, 533–567. MR**1888088**, DOI 10.1007/s002200100527 - Michael T. Anderson and Jeff Cheeger,
*Diffeomorphism finiteness for manifolds with Ricci curvature and $L^{n/2}$-norm of curvature bounded*, Geom. Funct. Anal.**1**(1991), no. 3, 231–252. MR**1118730**, DOI 10.1007/BF01896203 - Y. Fourès-Bruhat,
*Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires*, Acta Math.**88**(1952), 141–225 (French). MR**53338**, DOI 10.1007/BF02392131 - Jeff Cheeger,
*Finiteness theorems for Riemannian manifolds*, Amer. J. Math.**92**(1970), 61–74. MR**263092**, DOI 10.2307/2373498 - Demetrios Christodoulou and Sergiu Klainerman,
*The global nonlinear stability of the Minkowski space*, Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993. MR**1316662** - F. G. Friedlander,
*The wave equation on a curved space-time*, Cambridge Monographs on Mathematical Physics, No. 2, Cambridge University Press, Cambridge-New York-Melbourne, 1975. MR**0460898** - Gregory J. Galloway,
*Maximum principles for null hypersurfaces and null splitting theorems*, Ann. Henri Poincaré**1**(2000), no. 3, 543–567. MR**1777311**, DOI 10.1007/s000230050006 - S. W. Hawking and G. F. R. Ellis,
*The large scale structure of space-time*, Cambridge Monographs on Mathematical Physics, No. 1, Cambridge University Press, London-New York, 1973. MR**0424186**, DOI 10.1017/CBO9780511524646 - Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Marsden,
*Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity*, Arch. Rational Mech. Anal.**63**(1976), no. 3, 273–294 (1977). MR**420024**, DOI 10.1007/BF00251584 - Sergiu Klainerman and Igor Rodnianski,
*Causal geometry of Einstein-vacuum spacetimes with finite curvature flux*, Invent. Math.**159**(2005), no. 3, 437–529. MR**2125732**, DOI 10.1007/s00222-004-0365-4 - S. Klainerman and I. Rodnianski,
*A geometric approach to the Littlewood-Paley theory*, Geom. Funct. Anal.**16**(2006), no. 1, 126–163. MR**2221254**, DOI 10.1007/s00039-006-0551-1 - S. Klainerman and I. Rodnianski,
*Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux*, Geom. Funct. Anal.**16**(2006), no. 1, 164–229. MR**2221255**, DOI 10.1007/s00039-006-0557-8
[KR4]KR4 S. Klainerman and I. Rodnianski, - P. Petersen, S. D. Shteingold, and G. Wei,
*Comparison geometry with integral curvature bounds*, Geom. Funct. Anal.**7**(1997), no. 6, 1011–1030. MR**1487752**, DOI 10.1007/s000390050035 - Peter Petersen,
*Convergence theorems in Riemannian geometry*, Comparison geometry (Berkeley, CA, 1993–94) Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 167–202. MR**1452874**, DOI 10.2977/prims/1195166127
[Sob]Sob S. Sobolev,

*A Kirchoff-Sobolev parametrix for the wave equation in curved space-time*, preprint [KR5]KR5 S. Klainerman and I. Rodnianski,

*A large data break-down criterion in General Relativity*, submitted to J. Amer. Math. Soc.

*Methodes nouvelle a resoudre le probleme de Cauchy pour les equations lineaires hyperboliques normales*, Matematicheskii Sbornik, vol. 1 (43) 1936, 31-79. [Wang]Wang Q. Wang,

*Causal geometry of Einstein vacuum space-times.*Ph.D. thesis, Princeton University, 2006.

## Additional 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