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Abstract

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 analyze the case of a 2-dimensional symmetry group generated by two null vector fields. Its relevance to the classification of all the symmetric isolated horizons, including the non-extremal once, is explained.