Uniqueness of supersymmetric AdS5 black holes with SU(2) symmetry

We prove that any supersymmetric solution to five-dimensional minimal gauged supergravity with SU(2) symmetry, that is timelike outside an analytic horizon, is a Gutowski-Reall black hole or its near-horizon geometry. The proof combines a delicate near-horizon analysis with the general form for a Kähler metric with cohomogeneity-1 SU(2) symmetry. We also prove that any timelike supersymmetric soliton solution to this theory, with SU(2) symmetry and a nut or a complex bolt, has a Kähler base with enhanced U(1)×SU(2) symmetry, and we exhibit a family of asymptotically AdS5/Zp solitons for p≥3 with a bolt in this class.