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Maximal \({\mathsf {Sp}}(4,{\mathbb {R}})\) surface group representations, minimal immersions and cyclic surfaces

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Abstract

Let S be a closed surface of genus at least 2. For each maximal representation \(\rho : \pi _1(S){\rightarrow }{\mathsf {Sp}}(4,{\mathbb {R}})\) in one of the \(2g-3\) exceptional connected components, we prove there is a unique conformal structure on the surface in which the corresponding equivariant harmonic map to the symmetric space \({\mathsf {Sp}}(4,{\mathbb {R}})/{\mathsf {U}}(2)\) is a minimal immersion. Using a Higgs bundle parameterization of these components, we give a mapping class group invariant parameterization of such components as fiber bundles over Teichmüller space. Unlike Labourie’s recent results on Hitchin components, these bundles are not vector bundles.

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Acknowledgments

I would like to thank Daniele Alessandrini, Steve Bradlow, and François Labourie for many fruitful discussions. I am very grateful to Marco Spinaci for many enlightening email correspondences and useful comments. Also, I would like to thank Qiongling Li and Andy Sanders for countless stimulating conversations about representation varieties, harmonic maps and Higgs bundles. I acknowledge the support from US National Science Foundation Grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). I have benefited greatly from the opportunities the GEAR Network has provided me.

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  1. Department of Mathematics, University of Illinois, Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, USA

    Brian Collier

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  1. Brian Collier
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Collier, B. Maximal \({\mathsf {Sp}}(4,{\mathbb {R}})\) surface group representations, minimal immersions and cyclic surfaces. Geom Dedicata 180, 241–285 (2016). https://doi.org/10.1007/s10711-015-0101-9

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