Minimal surfaces for Hitchin representations

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  • Song Dai, Tianjin University, China
  • Qiongling Li, California Institute of Technology
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.
Original languageEnglish
JournalJournal of Differential Geometry
Pages (from-to)47-77
Number of pages31
Publication statusPublished - May 2019

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