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Higgs bundles, harmonic
maps and pleated surfaces
Andreas Ott, Jan Swoboda, Richard Wentworth and Michael Wolf |
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Geometry & Topology 28 (2024) 3135–3220
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Abstract |
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This paper unites the gauge-theoretic and hyperbolic-geometric perspectives on the asymptotic geometry of the character variety of representations of a surface group. Specifically, we find an asymptotic correspondence between the analytically defined limiting configuration of a sequence of solutions to the self-duality equations on a closed Riemann surface constructed by Mazzeo, Swoboda, Weiß and Witt, and the geometric topological shear-bend parameters of equivariant pleated surfaces in hyperbolic three-space due to Bonahon and Thurston. The geometric link comes from the nonabelian Hodge correspondence and a study of high-energy degenerations of harmonic maps. Our result has several applications. We prove: (1) the local invariance of the partial compactification of the moduli space of solutions to the self-duality equations by limiting configurations; (2) a refinement of the harmonic maps characterization of the Morgan–Shalen compactification of the character variety; and (3) a comparison between the family of complex projective structures defined by a quadratic differential and the realizations of the corresponding flat connections as Higgs bundles, as well as a determination of the asymptotic shear-bend cocycle of Thurston’s pleated surface. |
Keywords
$\mathrm{SL}(2,\mathbb{C})$–representations of surface
groups, Higgs bundles, nonabelian Hodge correspondence,
equivariant harmonic maps, pleated surfaces, self-duality
equations, complex projective structures
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Mathematical Subject Classification
Primary: 32G15, 53C07, 53C43
Secondary: 57K32
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References |
Publication
Received: 21 November 2021
Revised: 13 February 2023
Accepted: 6 May 2023
Published: 25 November 2024
Proposed: Simon Donaldson
Seconded: András I Stipsicz, Tomasz S Mrowka
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Authors |
| Andreas Ott | |
| Mathematisches Institut Ruprecht-Karls-Universität Heidelberg Heidelberg Germany |
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| Jan Swoboda | |
| Mathematisches Institut Ruprecht-Karls-Universität Heidelberg Heidelberg Germany |
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| Richard Wentworth | |
| Department of Mathematics University of Maryland College Park, MD United States |
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| Michael Wolf | |
| School of Mathematics Georgia Institute of Technology Atlanta, GA United States |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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