Geometry and Topology Seminar

Anti-de Sitter n-space is a complete Lorentzian manifold of constant sectional curvature -1 across all non-degenerate 2-planes. In dimension 3, anti-de Sitter space identifies with PSL_2(R), equipped with (a constant multiple of) its Killing metric, and the Lorentzian isometry group is (up to finite index) PSL_2(R) x PSL_2(R) acting via left and right multiplication. Recently attention has been devoted to understanding which subgroups of the isometry group admit properly discontinous actions, thus yielding what are called complete AdS 3-manifolds. I will describe aspects of this program and outline a construction of a new class of non-compact AdS 3-manifolds. The main technical tool in this case is the theory of equivariant harmonic maps.