Higgs bundles, isomonodromic leaves and minimal surfaces

In these lectures, I will present a gauge theoretical construction of the joint moduli space of Higgs bundles on a closed Riemann surface, where the Riemann surface structure is allowed to vary in the Teichmüller space of the underlying smooth surface. The nonabelian Hodge correspondence defines a foliation on the space whose holonomy recovers the natural mapping class group action on the character variety. This picture also has some strong implications for the space of equivariant minimal surfaces. This is based on a joint work with Brian Collier and Richard Wentworth.