Communications in Analysis and Geometry
Volume 29 (2021)
Stationary axisymmetric black holes with matter
Pages: 19 – 76
We provide a geometric framework for the construction of non-vacuum black holes whose metrics are stationary and axisymmetric. Under suitable assumptions we show that the Einstein equations reduce to an Einstein-harmonic map type system and analyze the compatibility of the resulting equations.
This framework is fundamental to our construction  of metric-stationary axisymmetric bifurcations of Kerr solving the Einstein–Klein–Gordon system, and as such, we include specializations of all of our formulas to the case of a time-periodic massive scalar field.
Received 29 October 2015
Accepted 23 May 2018
Published 11 March 2021