Uniqueness of Extremal Kerr and Kerr-Newman Black Holes

We prove that the only four dimensional, stationary, rotating, asymptotically flat (analytic) vacuum black hole with a single degenerate horizon is given by the extremal Kerr solution. We also prove a similar uniqueness theorem for the extremal Kerr-Newman solution. This closes a longstanding gap in the black hole uniqueness theorems.
Comment: 12 pages, v2: important references to earlier work added, v3: minor clarifications