Higgs bundles, harmonic maps, and pleated surfaces
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2020
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| _version_ | 1866916504627314688 |
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| author | Ott, Andreas Swoboda, Jan Wentworth, Richard Wolf, Michael |
| author_facet | Ott, Andreas Swoboda, Jan Wentworth, Richard Wolf, Michael |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2004_06071 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | Higgs bundles, harmonic maps, and pleated surfaces Ott, Andreas Swoboda, Jan Wentworth, Richard Wolf, Michael Differential Geometry Primary 32G15, 53C07, 53C43, Secondary 57K32 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. |
| title | Higgs bundles, harmonic maps, and pleated surfaces |
| topic | Differential Geometry Primary 32G15, 53C07, 53C43, Secondary 57K32 |
| url | https://arxiv.org/abs/2004.06071 |