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Minimal surfaces for Hitchin representations

Dai, Song and Li, Qiongling (2019) Minimal surfaces for Hitchin representations. Journal of Differential Geometry, 112 (1). pp. 47-77. ISSN 0022-040X. doi:10.4310/jdg/1557281006. https://resolver.caltech.edu/CaltechAUTHORS:20190523-145132149

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Abstract

Given a reductive representation ρ:π_1(S) → G, there exists a ρ-equivariant harmonic map f from the universal cover of a fixed Riemann surface Σ 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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.4310/jdg/1557281006DOIArticle
https://arxiv.org/abs/1605.09596arXivDiscussion Paper
Additional Information:© 2019 International Press. Received June 3, 2016. 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).
Funders:
Funding AgencyGrant Number
National Natural Science Foundation of China11601369
Danish National Research FoundationDNRF95
Issue or Number:1
DOI:10.4310/jdg/1557281006
Record Number:CaltechAUTHORS:20190523-145132149
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20190523-145132149
Official 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
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:95768
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:23 May 2019 22:16
Last Modified:16 Nov 2021 17:15

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