Extremal isolated horizons: a local uniqueness theorem

We derive all the axi-symmetric, vacuum and electrovac extremal isolated horizons. It turns out that for every horizon in this class, the induced metric tensor, the rotation 1-form potential and the pullback of the electromagnetic field necessarily coincide with those induced by the monopolar, extremal Kerr–Newman solution on the event horizon. We also discuss the general case of a symmetric, extremal isolated horizon. In particular, we analyse the case of a two-dimensional symmetry group generated by two null vector fields. Its relevance to the classification of all the symmetric isolated horizons, including the non-extremal ones, is explained.