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**Rigidity of stationary black holes with small angular momentum on the horizon.**
*(English)*
Zbl 1311.35309

Authors’ 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 by D. Sudarsky and R. M. Wald [“Mass formulas for stationary Einstein-Yang-Mills black holes and a simple proof of two staticity theorems”, Phys. Rev. D (3) 47, No. 12, R5209–R5213 (1993, doi:http://dx.doi.org/10.1103/PhysRevD.47.R5209)] 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.

The proof brings together ideas from our previous work with ideas from the classical work by D. Sudarsky and R. M. Wald [“Mass formulas for stationary Einstein-Yang-Mills black holes and a simple proof of two staticity theorems”, Phys. Rev. D (3) 47, No. 12, R5209–R5213 (1993, doi:http://dx.doi.org/10.1103/PhysRevD.47.R5209)] 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.

Reviewer: Anthony D. Osborne (Keele)

### MSC:

35Q75 | PDEs in connection with relativity and gravitational theory |

35L72 | Second-order quasilinear hyperbolic equations |

53Z05 | Applications of differential geometry to physics |

35Q76 | Einstein equations |

83C10 | Equations of motion in general relativity and gravitational theory |

83C57 | Black holes |