## Journal of Differential Geometry

### 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: the $q_n$ and $q_{n-1}$ cases. 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.

#### Note

The first author is supported by NSFC grant No. 11601369.

#### Note

The second author is supported in part by a grant from the Danish National Research Foundation (DNRF95).

#### Article information

Source
J. Differential Geom., Volume 112, Number 1 (2019), 47-77.

Dates
First available in Project Euclid: 8 May 2019

https://projecteuclid.org/euclid.jdg/1557281006

Digital Object Identifier
doi:10.4310/jdg/1557281006

Mathematical Reviews number (MathSciNet)
MR3948227

Zentralblatt MATH identifier
07054919

#### Citation

Dai, Song; Li, Qiongling. Minimal surfaces for Hitchin representations. J. Differential Geom. 112 (2019), no. 1, 47--77. doi:10.4310/jdg/1557281006. https://projecteuclid.org/euclid.jdg/1557281006