Higgs bundles, harmonic maps and pleated surfaces

Ott, Andreas¹ ; Swoboda, Jan¹ ; Wentworth, Richard² ; Wolf, Michael³

¹ Mathematisches Institut, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany
² Department of Mathematics, University of Maryland, College Park, MD, United States
³ School of Mathematics, Georgia Institute of Technology, Atlanta, GA, United States

Geometry & topology, Tome 28 (2024) no. 7, pp. 3135-3220.

Résumé

This paper unites the gauge-theoretic and hyperbolic-geometric perspectives on the asymptotic geometry of the character variety of $\mathrm{SL}(2,ℂ)$ 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 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.

Détail

DOI : 10.2140/gt.2024.28.3135

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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