Natural maps in higher Teichmüller theory

In this thesis we consider harmonic maps and barycentric maps in the context of higher Teichmüller theory. We are particularly interested in how these maps can be used to study Hitchin representations. The main results of this work are as follows. Our first result states that equivariant harmonic maps into non-compact symmetric spaces that satisfy suitable non-degeneracy conditions depend in a real analytic fashion on the metric of the domain manifold and the representations they are associated to. For our second result we consider the energy functional on Teichmüller space that is associated to a Hitchin representation. We prove that this functional is strictly plurisubharmonic for Hitchin representations into either PSL(n, R), PSp(2n, R), PSO(n, n + 1) or G_2. In the third part of this thesis we examine the energy functional on Teichmüller space that is associated to a metric on a surface. We prove that the simple length spectrum of a non-positively curved metric is determined by its energy functional. We use this to prove that hyperbolic metrics and singular flat metrics induced by quadratic differentials are determined, up to isotopy, by their energy functional. Our next result concerns the harmonic heat flow for maps from a compact Riemannian manifold into a Riemannian manifold of non-positive curvature. We prove that if the harmonic heat flow converges to a harmonic map that is a non-degenerate critical point of the Dirichlet energy, then it converges exponentially fast. In the final part of this thesis we study the barycenter construction of Besson–Courtois–Gallot. We prove that for any Fuchsian representation and Hitchin representation into SL(n, R) there exists a natural map from the hyperbolic plane to SL(n, R)/SO(n) that intertwines the actions of the two representations. We put these maps forward as a new way to parametrise and study Hitchin components