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A key result in the proof of black hole uniqueness in 4-dimensions is that astationary black hole that is ``rotating''--i.e., is such that the stationaryKilling field is not everywhere normal to the horizon--must be axisymmetric.The proof of this result in 4-dimensions relies on the fact that the orbits ofthe stationary Killing field on the horizon have the property that they mustreturn to the same null geodesic generator of the horizon after a certainperiod, $P$. This latter property follows, in turn, from the fact that thecross-sections of the horizon are two-dimensional spheres. However, inspacetimes of dimension greater than 4, it is no longer true that the orbits ofthe stationary Killing field on the horizon must return to the same nullgeodesic generator. In this paper, we prove that, nevertheless, a higherdimensional stationary black hole that is rotating must be axisymmetric. Noassumptions are made concerning the topology of the horizon cross-sectionsother than that they are compact. However, we assume that the horizon isnon-degenerate and, as in the 4-dimensional proof, that the spacetime isanalytic.Comment: 24 pages, no figures, v2: footnotes and references added, v3:  numerous minor revision

A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric