R} \times \{ {\Bbb R}^4 \setminus {\Bbb B}^4 \}$, then the black hole spacetime is uniquely characterized by its mass, electric charge, and two independent angular momenta, and hence must be isometric to the Cveti[č]{}-Youm solution with equal charges.* ]{}

Before presenting our proof, we would like to make a few comments concerning the assumptions made in our theorem. In order to obtain global results, we need the symmetry-condition (1), which, in particular makes it possible to reduce five-dimensional minimal supergravity to a non-linear sigma model with certain symmetries as shown in [@MO; @MO2]. Since all known exact black hole solutions in higher dimensions admit multiple axial isometries, our additional symmetry-condition (1) does not appear to be too restrictive. However, we should note that the rigidity theorem [@HIW] (see also [@MI08]) in higher dimensions—which is recently shown to be applicable also to EMCS theory [@HI]—only guarantees the existence of a single rotational isometry (provided the spacetime metric and other fields are real, analytic), and therefore at present, the condition (1) is not yet fully justified. In this respect, note also that the possibility for higher-dimensional black holes with fewer isometries than ${\Bbb R} \times U(1)\times U(1)$ has been suggested [@Reall03]. 