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.

中文翻译：

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Hitchin 表示的最小曲面

给定一个还原表示 $\rho:\pi_1(S)\rightarrow G$，存在一个 $\rho$-等变调和映射 $f$ 从固定黎曼曲面 $\Sigma$ 的普遍覆盖到对称空间 $ G/K$ 关联到 $G$。如果 $f$ 的 Hopf 微分消失，则谐波映射最小。在本文中，我们研究了与 Hitchin 分量的子位点相关的对称空间内浸没最小表面的性质：$q_n$ 和 $q_{n-1}$ case。首先，我们表明最小表面的回拉度量在同一保形类中支配双曲度量的常数倍，并且具有很强的刚性属性。其次，我们表明浸入的最小表面永远不会与对称空间内的任何平面相切。作为直接推论，最小表面的回拉度量总是严格负弯曲的。最后，我们找到了一个完全解耦的系统来近似耦合的 Hitchin 系统。