On the local extension of Killing vector-fields in Ricci flat manifolds

Authors:
Alexandru D. Ionescu and Sergiu Klainerman

Journal:
J. Amer. Math. Soc. **26** (2013), 563-593

MSC (2010):
Primary 53B30, 83C05, 83C57

Published electronically:
November 14, 2012

MathSciNet review:
3011421

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We revisit the extension problem for Killing vector-fields in smooth Ricci flat manifolds, and its relevance to the black hole rigidity problem. We prove both a stronger version of the main local extension result established earlier, as well as two types of results concerning non-extendibility. In particular, we show that one can find local, stationary, vacuum extensions of a Kerr solution , , in a future neighborhood of any point of the past horizon lying outside both the bifurcation sphere and the axis of symmetry, which admit no extension of the Hawking vector-field of . This result illustrates one of the major difficulties one faces in trying to extend Hawking's rigidity result to the more realistic setting of smooth stationary solutions of the Einstein vacuum equations; unlike in the analytic situation, one cannot hope to construct an additional symmetry of stationary solutions (as in Hawking's Rigidity Theorem) by relying only on local information.

**[1]**Spyros Alexakis, Alexandru D. Ionescu, and Sergiu Klainerman,*Hawking’s local rigidity theorem without analyticity*, Geom. Funct. Anal.**20**(2010), no. 4, 845–869. MR**2729279**, 10.1007/s00039-010-0082-7**[2]**S. Alexakis, A. D. Ionescu, and S. Klainerman,*Uniqueness of smooth stationary black holes in vacuum: small perturbations of the Kerr spaces*, Comm. Math. Phys.**299**(2010), no. 1, 89–127. MR**2672799**, 10.1007/s00220-010-1072-1**[3]**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****[4]**Piotr T. Chruściel,*On rigidity of analytic black holes*, Comm. Math. Phys.**189**(1997), no. 1, 1–7. MR**1478527**, 10.1007/s002200050187**[5]**S. W. Hawking and G. F. R. Ellis,*The large scale structure of space-time*, Cambridge University Press, London-New York, 1973. Cambridge Monographs on Mathematical Physics, No. 1. MR**0424186****[6]**Lars Hörmander,*The analysis of linear partial differential operators. IV*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 275, Springer-Verlag, Berlin, 1985. Fourier integral operators. MR**781537****[7]**Alexandru D. Ionescu and Sergiu Klainerman,*On the uniqueness of smooth, stationary black holes in vacuum*, Invent. Math.**175**(2009), no. 1, 35–102. MR**2461426**, 10.1007/s00222-008-0146-6**[8]**Alexandru D. Ionescu and Sergiu Klainerman,*Uniqueness results for ill-posed characteristic problems in curved space-times*, Comm. Math. Phys.**285**(2009), no. 3, 873–900. MR**2470908**, 10.1007/s00220-008-0650-y- [9] J. Luk, On the local existence for the characteristic initial value problem in general relativity, Preprint (2011).
**[10]**Katsumi Nomizu,*On local and global existence of Killing vector fields*, Ann. of Math. (2)**72**(1960), 105–120. MR**0119172****[11]**A. D. Rendall,*Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations*, Proc. Roy. Soc. London Ser. A**427**(1990), no. 1872, 221–239. MR**1032984****[12]**Gilbert Weinstein,*On rotating black holes in equilibrium in general relativity*, Comm. Pure Appl. Math.**43**(1990), no. 7, 903–948. MR**1072397**, 10.1002/cpa.3160430705

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (2010):
53B30,
83C05,
83C57

Retrieve articles in all journals with MSC (2010): 53B30, 83C05, 83C57

Additional Information

**Alexandru D. Ionescu**

Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Email:
aionescu@math.princeton.edu

**Sergiu Klainerman**

Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544

Email:
seri@math.princeton.edu

DOI:
http://dx.doi.org/10.1090/S0894-0347-2012-00754-1

Received by editor(s):
August 22, 2011

Received by editor(s) in revised form:
August 18, 2012

Published electronically:
November 14, 2012

Additional Notes:
The first author was supported in part by a Packard fellowship.

The second author was supported in part by NSF grant 0601186 as well as by the Fondation des Sciences Mathématiques de Paris.

Both authors were also supported in part by NSF-FRG grant DMS-1065710.

Article copyright:
© Copyright 2012
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.