Abstract

In the first part of the talk I will present uniqueness theorems for asymptotically flat, stationary, extremal, vacuum black hole solutions, in four and five dimensions with one and two commuting rotational Killing fields respectively. As in the non-extremal case, these problems may be cast as boundary value problems on the two dimensional orbit space. In four dimensions this allows us to establish the uniqueness of extremal Kerr amongst asymptotically flat, stationary, rotating, vacuum black holes with a single extremal horizon. In the second part of the talk, I will consider minimal supergravity restricted to stationary solutions with two commuting rotational Killing fields. I will show, using the non-linear sigma model description, that for this class of solutions this theory is completely integrable by constructing the corresponding Lax pair. Finally I will discuss how one may construct an inverse scattering method for this theory.