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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

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
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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 $ \mathcal {K}(m,a)$, $ 0<a<m$, in a future neighborhood of any point $ p$ 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 $ \mathcal {K}(m,a)$. 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.


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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
PII: S 0894-0347(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.