Seminarium Teorii Względności i Grawitacji

The Einstein equations for a spacetime containing an extremal horizon induce a set of equations determined completely by quantities intrinsic to a compact cross-section of the horizon. Significant progress on classifying solutions to these equations was recently made by Dunajski and Lucietti, who showed that any non-static vacuum solution must admit a Killing vector field.I will present joint work with David Katona and James Lucietti extending the Dunajski-Lucietti proof to four-dimensional Einstein-Maxwell theory. It follows that any non-trivial cross-section in this theory is given by the extremal Kerr-Newman family. I will also discuss rigidity results for the quasi-Einstein equation and a more general extremal horizon equation introduced by Kaminski and Lewandowski.