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We make use of the black hole holograph construction of [I. R\'acz,Stationary black holes as holographs, Class. Quantum Grav. 31, 035006 (2014)]to analyse the existence of Killing spinors in the domain of dependence of thehorizons of distorted black holes. In particular, we provide conditions on thebifurcation sphere ensuring the existence of a Killing spinor. These conditionscan be understood as restrictions on the curvature of the bifurcation sphereand ensure the existence of an axial Killing vector on the 2-surface. We obtainthe most general 2-dimensional metric on the bifurcation sphere for which thesecurvature conditions are satisfied. Remarkably, these conditions are found tobe so restrictive that, in the considered particular case, the free data on thebifurcation surface (determining a distorted black hole spacetime) iscompletely determined by them. In addition, we formulate further conditions onthe bifurcation sphere ensuring that the Killing vector associated to theKilling spinor is Hermitian. Once the existence of a Hermitian Killing vectoris guaranteed, one can use a characterisation of the Kerr spacetime due to Marsto identify the particular subfamily of 2-metrics giving rise to a member ofthe Kerr family in the black hole holograph construction. Our analysis shedslight on the role of asymptotic flatness and curvature conditions on thebifurcation sphere in the context of the problem of uniqueness of stationaryblack holes. The Petrov type of the considered distorted black hole spacetimesis also determined.

Killing spinor data on distorted black hole horizons and the uniqueness of stationary vacuum black holes