Duke Mathematical Journal

Rigidity of stationary black holes with small angular momentum on the horizon

S. Alexakis, A. D. Ionescu, and S. Klainerman

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We prove a black hole rigidity result for slowly rotating stationary solutions of the Einstein vacuum equations. More precisely, we prove that the domain of outer communications of a regular stationary vacuum is isometric to the domain of outer communications of a Kerr solution, provided that the stationary Killing vector-field T is small (depending only on suitable regularity properties of the black hole) on the bifurcation sphere. No other global restrictions are necessary.

The proof brings together ideas from our previous work with ideas from the classical work of Sudarsky and Wald on the staticity of stationary black hole solutions with zero angular momentum on the horizon. It is thus the first uniqueness result, in the framework of smooth, asymptotically flat, stationary solutions, which combines local considerations near the horizon, via Carleman estimates, with information obtained by global elliptic estimates.

Article information

Duke Math. J., Volume 163, Number 14 (2014), 2603-2615.

First available in Project Euclid: 31 October 2014

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Zentralblatt MATH identifier

Primary: 35L72: Quasilinear second-order hyperbolic equations
Secondary: 53Z05: Applications to physics

Einstein black holes rigidity


Alexakis, S.; Ionescu, A. D.; Klainerman, S. Rigidity of stationary black holes with small angular momentum on the horizon. Duke Math. J. 163 (2014), no. 14, 2603--2615. doi:10.1215/00127094-2819517. https://projecteuclid.org/euclid.dmj/1414762065

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