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May 2019 Minimal surfaces for Hitchin representations
Song Dai, Qiongling Li
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J. Differential Geom. 112(1): 47-77 (May 2019). DOI: 10.4310/jdg/1557281006

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.

Funding Statement

The first author is supported by NSFC grant No. 11601369.
The second author is supported in part by a grant from the Danish National Research Foundation (DNRF95).

Citation

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Song Dai. Qiongling Li. "Minimal surfaces for Hitchin representations." J. Differential Geom. 112 (1) 47 - 77, May 2019. https://doi.org/10.4310/jdg/1557281006

Information

Received: 3 June 2016; Published: May 2019
First available in Project Euclid: 8 May 2019

zbMATH: 07054919
MathSciNet: MR3948227
Digital Object Identifier: 10.4310/jdg/1557281006

Rights: Copyright © 2019 Lehigh University

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Vol.112 • No. 1 • May 2019
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