Minimal surfaces for Hitchin representations

TL;DR

This paper studies minimal surfaces associated with Hitchin representations, revealing their metric properties, rigidity, and curvature, and provides a new approach to approximate the Hitchin system.

Contribution

It characterizes the metric dominance, rigidity, and curvature of minimal surfaces in Hitchin components and introduces a decoupled system to approximate the Hitchin equations.

Findings

Pullback metric dominates hyperbolic metric in the same conformal class.

Minimal surfaces are never tangent to flats in the symmetric space.

Pullback metric is always strictly negatively curved.

Abstract

Given a reductive representation $\rho: \pi_1(S)\rightarrow G$, there exists a $\rho$-equivariant harmonic map $f$ from the universal cover of a fixed Riemann surface $\Sigma$ to the symmetric space $G/K$ associated to $G$. If the Hopf differential of $f$ vanishes, the harmonic map is then minimal. In this paper, we investigate the properties of immersed minimal surfaces inside symmetric space associated to a subloci of Hitchin component: $q_n$ and $q_{n-1}$ case. First, we show that the pullback metric of the minimal surface dominates a constant multiple of the hyperbolic metric in the same conformal class and has a strong rigidity property. Secondly, we show that the immersed minimal surface is never tangential to any flat inside the symmetric space. As a direct corollary, the pullback metric of the minimal surface is always strictly negatively curved. In the end, we find a fully decoupled system to approximate the coupled Hitchin system.