Uniqueness of Extremal Kerr and Kerr-Newman Black Holes

TL;DR

The paper addresses the gap in four-dimensional black hole uniqueness by proving that the only stationary, rotating, asymptotically flat analytic vacuum black hole with a single degenerate horizon is the extremal Kerr solution, and that the analogous extremal Kerr-Newman solution is unique in Einstein-Maxwell theory. The authors combine near-horizon extremal geometry results (NHEK, extremal electrovac throats) with Mazur-type harmonic map methods in Weyl-Papapetrou and Ernst-potential formalisms, establishing a Mazur identity-based argument that yields a globally bounded quantity sigma whose vanishing implies metric and field equivalence. This leads to direct global uniqueness proofs (Theorems 1 and 2) for extremal Kerr and extremal Kerr-Newman, respectively. By closing this longstanding gap, the work strengthens the foundation of black hole uniqueness and informs related topics such as extremal horizon thermodynamics and holographic correspondences.

Abstract

We prove that the only four dimensional, stationary, rotating, asymptotically flat (analytic) vacuum black hole with a single degenerate horizon is given by the extremal Kerr solution. We also prove a similar uniqueness theorem for the extremal Kerr-Newman solution. This closes a longstanding gap in the black hole uniqueness theorems.