Higgs bundles, harmonic maps, and pleated surfaces
This paper unites the gauge-theoretic and hyperbolic-geometric perspectives on the asymptotic geometry of the character variety of SL(2,C) representations of a surface group. Specifically, we find an asymptotic correspondence between the analytically defined limiting configuration of a sequence of s...
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| Main Authors: | , , , |
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| Format: | Article (Journal) Chapter/Article |
| Language: | English |
| Published: |
12 Feb 2021
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| Edition: | Version v3 |
| In: |
Arxiv
Year: 2021, Pages: 1-77 |
| DOI: | 10.48550/arXiv.2004.06071 |
| Online Access: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.48550/arXiv.2004.06071 Verlag, lizenzpflichtig, Volltext: http://arxiv.org/abs/2004.06071 
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| Author Notes: | Andreas Ott, Jan Swoboda, Richard Wentworth, and Michael Wolf |
| Summary: | This paper unites the gauge-theoretic and hyperbolic-geometric perspectives on the asymptotic geometry of the character variety of SL(2,C) representations of a surface group. Specifically, we find an asymptotic correspondence between the analytically defined limiting configuration of a sequence of solutions to the SU(2) self-duality equations on a closed Riemann surface constructed by Mazzeo-Swoboda-Weiss-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. |
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| Item Description: | Version 1 vom 13. April 2020, Version 2 vom 6. Juli 2020, Version 3 vom 12. Februar 2021 Gesehen am 07.10.2022 |
| Physical Description: | Online Resource |
| DOI: | 10.48550/arXiv.2004.06071 |