Harmonic maps for Hitchin representations

Abstract

Let (S,g₀) be a hyperbolic surface, ${\rho}$ be a Hitchin representation for ${PSL(n,{\mathbb{R}})}$, and f be the unique ${\rho}$-equivariant harmonic map from ${({\widetilde{S}}, \widetilde g_0)}$ to the corresponding symmetric space. We show its energy density satisfies ${e(f)\geq 1}$ and equality holds at one point only if ${e(f)\equiv 1}$ and ${\rho}$ is the base ${n}$-Fuchsian representation of (S,g₀). In particular, we show given a Hitchin representation ${\rho}$ for ${PSL(n,{\mathbb{R}})}$, every ${\rho}$-equivariant minimal immersion f from the hyperbolic plane ${{\mathbb{H}}^2}$ into the corresponding symmetric space X is distance-increasing, i.e. ${f^*g_{X}\geq g_{{\mathbb{H}}^2}}$. Equality holds at one point only if it holds everywhere and ${\rho}$ is an n-Fuchsian representation.