A boundary value problem for the five-dimensional stationary rotating black holes

TL;DR

This work addresses uniqueness of five-dimensional stationary rotating vacuum black holes with spherical horizon topology by recasting Einstein equations as a nonlinear sigma-model on $SL(3,\mathbb{R})/SO(3)$ and applying the Mazur identity. By enforcing asymptotic flatness and regularity on the rotation planes and horizon, it proves that the solutions are uniquely fixed by the mass $M$ and two angular momenta $(J_\phi,J_\psi)$, corresponding to the Myers-Perry family in this class. The work clarifies how horizon topology constrains uniqueness and discusses why the Mazur-identity-based approach cannot be extended straightforwardly to higher dimensions. It also highlights the existence of nonunique solutions with different horizon topology, such as black rings, within the broader five-dimensional landscape.

Abstract

We study the boundary value problem for the stationary rotating black hole solutions to the five-dimensional vacuum Einstein equation. Assuming the two commuting rotational symmetry and the sphericity of the horizon topology, we show that the black hole is uniquely characterized by the mass, and a pair of the angular momenta.