On rigidity of stationary black holes under a nontrapping assumption

Abstract

We prove that a smooth, regular, asymptotically flat stationary vacuum black hole has a nonextremal Kerr exterior if no null geodesic orthogonal to its stationary Killing field is trapped, establishing the rigidity conjecture of Alexakis, Ionescu, and Klainerman. In the rotating case, we start with the rotational Killing field constructed near the bifurcation sphere and study a maximal domain carrying its complete circle action. The main difficulty is to recover boundary regularity and the curvature condition needed for local Killing extension. We derive uniform estimates along complete rotational orbits from nontrapping and combine them with convexity along short null geodesics. These estimates show that a nonempty remaining boundary in the ergoregion is a timelike photon surface for the stationary quotient metric. Its null geodesics lift to spacetime null geodesics orthogonal to the stationary field. The vacuum circularity identities give their continuation through contact with the ergosurface, keeping their stationary projections in a compact set. This contradicts nontrapping and yields the global rotational symmetry required by black hole uniqueness.