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On cyclic Higgs bundles

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Abstract

In this paper, we derive a maximum principle for a type of elliptic systems and apply it to analyze the Hitchin equation for cyclic Higgs bundles. We show several domination results on the pullback metric of the (possibly branched) minimal immersion f associated to cyclic Higgs bundles. Also, we obtain a lower and upper bound of the extrinsic curvature of the image of f. As an application, we give a complete picture for maximal \(Sp(4,{\mathbb {R}})\)-representations in the \(2g-3\) Gothen components and the Hitchin components.

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Acknowledgements

Qiongling Li wishes to thank Nicolas Tholozan for suggesting the problem of looking for a lower bound for the extrinsic curvature of the harmonic map. The authors acknowledge support from U.S. National Science Foundation Grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network).

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Authors and Affiliations

  1. Center for Applied Mathematics of Tianjin University, Tianjin University, No. 92 Weijinlu Nankai District, Tianjin, 300072, People’s Republic of China

    Song Dai

  2. Chern Institute of Mathematics, Nankai University, No. 94 Weijinlu Nankai District, Tianjin, 300071, People’s Republic of China

    Qiongling Li

Authors
  1. Song Dai
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  2. Qiongling Li
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Correspondence to Qiongling Li.

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Communicated by Ngaiming Mok.

Song Dai was supported by NSFC Grant no. 11601369. Qiongling Li was supported in part by the center of excellence grant ‘Center for Quantum Geometry of Moduli Spaces’ from the Danish National Research Foundation (DNRF95).

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Dai, S., Li, Q. On cyclic Higgs bundles. Math. Ann. 376, 1225–1260 (2020). https://doi.org/10.1007/s00208-018-1779-4

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  • DOI: https://doi.org/10.1007/s00208-018-1779-4

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