Minimal surfaces for Hitchin representations
Minimal surfaces for Hitchin representations
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DOI：
10.4310/jdg/1557281006
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发表时间：
2016-05
影响因子：
2.5
通讯作者：
S. Dai;Qiongling Li
中科院分区：
文献类型：
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作者：
S. Dai;Qiongling Li
给定一个约化表示\(\rho:\pi_1(S)\to G\)，存在一个从固定黎曼曲面\(\Sigma\)的万有覆盖到与\(G\)相关联的对称空间\(G/K\)的\(\rho\) - 等变调和映射\(f\)。如果\(f\)的霍普夫微分消失，那么该调和映射是极小的。在本文中，我们研究与希钦分量的一个子轨迹相关联的对称空间内浸入极小曲面的性质：\(q_n\)和\(q_{n - 1}\)情形。首先，我们证明极小曲面的拉回度量在相同共形类中控制双曲度量的某个常数倍，并且具有很强的刚性性质。其次，我们证明浸入极小曲面在对称空间内永远不会与任何平坦子空间相切。作为直接推论，极小曲面的拉回度量总是严格负曲率的。最后，我们找到一个完全解耦的系统来逼近耦合的希钦系统。
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.