Minimal surfaces for Hitchin representations

Abstract

Given a reductive representation $\rho :{\pi }_1(S)\to G$, 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.

24 pages, revised version, to appear in J. Differ. Geom

References in corpus (4)

• Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces
• G2 geometry and integrable systems
• Entropy degeneration of convex projective surfaces
• Hitchin harmonic maps are immersions