Surfaces with harmonic Gauss map in anti-de Sitter 3-space

Harmonic maps (or wave maps) from a Lorentzian surface into a 2-dimesional space form are associated with various special problems in geometry. In this thesis we study timelike and spacelike surfaces in the anti de Sitter 3-space that have harmonic maps with respect to either the first or the second fundamental form. In particular we show that timelike and spacelike surfaces with constant negative extrinsic curvature (and their associated parallel surfaces) are a natural example of this. We show that the theory of loop groups can be used to represent such surfaces, and we show how to solve the Cauchy problem for the spacelike case.