Harmonic maps for Hitchin representations

Abstract

Let $(S,g_0)$ be a hyperbolic surface, $\rho$ be a Hitchin representation for $PSL(n,\mathbb{R})$, and $f$ be the unique $\rho$-equivariant harmonic map from $(\tilde{S},{\tilde{g}}_0)$ to the corresponding symmetric space. We show its energy density satisfies $e(f)\ge 1$ and equality holds at one point only if $e(f)\equiv 1$ and $\rho$ is the base $n$-Fuchsian representation of $(S,g_0)$. In particular, we show given a Hitchin representation $\rho$ for $PSL(n,\mathbb{R})$, every $\rho$-equivariant minimal immersion $f$ from a hyperbolic plane ${\mathbb{H}}^2$ into the corresponding symmetric space $X$ is distance-increasing, i.e. $f^{\ast }(g_X)\ge g_{{\mathbb{H}}^2}$. Equality holds at one point only if it holds everywhere and $\rho$ is an $n$-Fuchsian representation.

14 pages, comments are welcome