Deligne-Hitchin moduli space and harmonic maps into hyperbolic 3-space passing through the sphere at infinity

The aim of the talk is to explain a construction of certain harmonic maps into two copies of hyperbolic 3-space considered as half spheres in $S^3$ with the equatorial $S^2$ as the boundary at infinity for both hyperbolic spaces. We will also explain how these maps are related to certain sections of the Deligne–Hitchin moduli space of flat $\lambda-SL(2, C)$ connections and how this aids understanding the geometry of this space, which has many interesting properties, such as a natural hyperkähler structure.