The Petrov type D equation on genus >0 sections of isolated horizons

The Petrov type D equation imposed on the 2-metric tensor and the rotation scalar of a cross-section of an isolated horizon can be used to uniquely distinguish the Kerr-(anti) de Sitter spacetime in the case the topology of the cross-section is that of a sphere. In the current paper we study that equation on closed 2-dimensional surfaces that have genus >0. We derive all the solutions assuming the embeddability in 4-dimensional spacetime that satisfies the vacuum Einstein equations with (possibly 0) cosmological constant. We prove all of them have constant Gauss curvature and zero rotation. Consequently, we provide a quasi-local argument for a black hole in 4-dimensional spacetime to have a topologically spherical cross-section.