## Duke Mathematical Journal

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

#### Abstract

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 $\mathbf{T}$ 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

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

Dates
First available in Project Euclid: 31 October 2014

https://projecteuclid.org/euclid.dmj/1414762065

Digital Object Identifier
doi:10.1215/00127094-2819517

Mathematical Reviews number (MathSciNet)
MR3273578

Zentralblatt MATH identifier
1311.35309

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

Keywords
Einstein black holes rigidity

#### Citation

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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