Harmonic maps for Hitchin representations

TL;DR

This paper investigates harmonic maps associated with Hitchin representations, establishing energy bounds and conditions for equality, and demonstrating that such maps are distance-increasing immersions with unique cases of equality.

Contribution

It proves energy density bounds for harmonic maps from hyperbolic surfaces to symmetric spaces and characterizes when equality occurs, linking to Fuchsian representations.

Findings

Energy density of harmonic maps is at least 1.

Equality in energy density occurs only for base n-Fuchsian representations.

Hitchin representations induce distance-increasing minimal immersions.

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 $(\widetilde S, \widetilde g_0)$ 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 $(S,g_0)$. 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. $f^*(g_{X})\geq g_{\mathbb H^2}$. Equality holds at one point only if it holds everywhere and $\rho$ is an $n$-Fuchsian representation.