Geometriae Dedicata

, Volume 180, Issue 1, pp 241–285 | Cite as

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.

Keywords

Character variety Higgs bundles Harmonic maps  Mapping class group Maximal representation 
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Notes

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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Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Illinois, Urbana-ChampaignUrbanaUSA

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