Metadata
Abstract
For every integer $g \ge 2$ we show the existence of a compact Riemann surface $\Sigma $ of genus $g$ such that the rank two trivial holomorphic vector bundle $\mathcal{O}^{\oplus 2}_{\Sigma }$ admits holomorphic connections with $\operatorname{SL}(2,\mathbb{R})$ monodromy and maximal Euler class. Such a monodromy representation is known to coincide with the Fuchsian uniformizing representation for some Riemann surface of genus $g$. This also answers a question of Calsamiglia, Deroin, Heu and Loray. The construction carries over to all very stable and compatible real holomorphic structures over the topologically trivial rank two bundle on $\Sigma $, and gives the existence of holomorphic connections with Fuchsian monodromy in these cases as well.
References
[AGM11] Thermodynamic bubble ansatz, J. High Energy Phys., Volume 2011 (2011) no. 9, 32, 46 pages | MR | DOI | Zbl
[AL18] AdS 3-manifolds and Higgs bundles, Proc. Am. Math. Soc., Volume 146 (2018) no. 2, pp. 845-860 | MR | DOI | Zbl
[BDH21] Irreducible flat -connections on the trivial holomorphic bundle, J. Math. Pures Appl. (9), Volume 149 (2021), pp. 28-46 | MR | DOI | Zbl
[BHS21] Constant mean curvature surfaces based on fundamental quadrilaterals, Math. Phys. Anal. Geom., Volume 24 (2021) no. 4, 37, 46 pages | MR | DOI | Zbl
[BIW10] Surface group representations with maximal Toledo invariant, Ann. Math. (2), Volume 172 (2010) no. 1, pp. 517-566 | DOI | MR | Zbl
[CDF14] Branched projective structures with Fuchsian monodromy, Geom. Topol., Volume 18 (2014), pp. 379-446 | MR | DOI | Zbl
[CDHL19] The Riemann–Hilbert mapping for systems over genus two curves, Bull. Soc. Math. Fr., Volume 147 (2019) no. 1, pp. 159-195 | DOI | Zbl
[Del70] Équations différentielles à points singuliers réguliers, Lecture Notes in Mathematics, 163, Springer, 1970 | MR | DOI | Zbl
[DM93] Commensurabilities among lattices in , Annals of Mathematics Studies, 132, Princeton University Press, 1993 | MR | DOI | Zbl
[Fal83] Real projective structures on Riemann surfaces, Compos. Math., Volume 48 (1983), pp. 223-269 | MR | Numdam | Zbl
[FGT16] Surface operators and separation of variables, J. High Energy Phys., Volume 2016 (2016) no. 1, 179, 54 pages | MR | DOI | Zbl
[Ghy95] Déformations des structures complexes sur les espaces homogènes de , J. Reine Angew. Math., Volume 468 (1995), pp. 113-138 | MR | DOI | Zbl
[GKM00] The monodromy groups of Schwarzian equations on closed Riemann surfaces, Ann. Math. (2), Volume 151 (2000) no. 2, pp. 625-704 | DOI | Zbl
[GMN13] Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math., Volume 234 (2013), pp. 239-403 | MR | DOI | Zbl
[Gol87] Projective structures with Fuchsian holonomy, J. Differ. Geom., Volume 25 (1987), pp. 297-326 | MR | DOI | Zbl
[Gol88] Topological components of spaces of representations, Invent. Math., Volume 93 (1988) no. 3, pp. 557-607 | MR | DOI | Zbl
[Gol03] The modular group action on real -characters of a one-holed torus, Geom. Topol., Volume 7 (2003), pp. 443-486 | MR | DOI | Zbl
[Gun67] Lectures on vector bundles over Riemann surfaces. Preliminary informal notes of university courses and seminars in mathematics, Mathematical Notes (Princeton), Princeton University Press, 1967 | MR | Zbl
[GW12] Anosov representations: domains of discontinuity and applications, Invent. Math., Volume 190 (2012) no. 2, pp. 357-438 | MR | DOI | Zbl
[Hau20] Ein Satz über die Abelschen Integrale 1. Gattung, Math. Z., Volume 6 (1920), pp. 219-237 | MR | Zbl | DOI
[Hej75] Monodromy groups and linearly polymorphic functions, Acta Math., Volume 135 (1975), pp. 1-55 | MR | DOI | Zbl
[Hel14] A spectral curve approach to Lawson symmetric CMC surfaces of genus 2, Math. Ann., Volume 360 (2014) no. 3-4, pp. 607-652 | MR | DOI | Zbl
[HH16] Abelianization of Fuchsian systems on a 4-punctured sphere and applications, J. Symplectic Geom., Volume 14 (2016) no. 4, pp. 1059-1088 | DOI | Zbl
[HHS18] Navigating the space of symmetric CMC surfaces, J. Differ. Geom., Volume 110 (2018) no. 3, pp. 413-455 | MR | DOI | Zbl
[Hit87] The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc. (3), Volume 55 (1987), pp. 59-126 | MR | DOI | Zbl
[Hit90] Harmonic maps from a 2-torus to the 3-sphere, J. Differ. Geom., Volume 31 (1990) no. 3, pp. 627-710 | MR | DOI | Zbl
[Kat76] An overview of Deligne’s work on Hilbert’s twenty-first problem, Mathematical developments arising from Hilbert problems (Proceedings of Symposia in Pure Mathematics), Volume 28, American Mathematical Society, 1976, pp. 537-557 | DOI | MR | Zbl
[KNPS15] Harmonic maps to buildings and singular perturbation theory, Commun. Math. Phys., Volume 336 (2015) no. 2, pp. 853-903 | MR | DOI | Zbl
[Lab06] Anosov flows, surface groups and curves in projective space, Invent. Math., Volume 165 (2006) no. 1, pp. 51-114 | MR | DOI | Zbl
[Law70] Complete minimal surfaces in , Ann. Math. (2), Volume 92 (1970), pp. 335-374 | MR | DOI | Zbl
[LS15] Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line, Int. Math. Res. Not., Volume 2015 (2015) no. 4, pp. 995-1043 | MR | DOI | Zbl
[Mas69] On a class of Kleinian groups., Ann. Acad. Sci. Fenn., Ser. A I, Volume 442 (1969), p. 8 | MR | Zbl
[Mil58] On the existence of a connection with curvature zero, Comment. Math. Helv., Volume 32 (1958), pp. 215-223 | MR | DOI | Zbl
[Moc16] Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces, J. Topol., Volume 9 (2016) no. 4, pp. 1021-1073 | MR | DOI | Zbl
[MS80] Moduli of vector bundles on curves with parabolic structures, Math. Ann., Volume 248 (1980), pp. 205-239 | MR | DOI | Zbl
[MY92] Moduli of parabolic stable sheaves, Math. Ann., Volume 293 (1992) no. 1, pp. 77-99 | MR | DOI | Zbl
[Oht82] A residue formula for Chern classes associated with logarithmic connections, Tokyo J. Math., Volume 5 (1982), pp. 13-21 | Zbl | MR | DOI
[SG16] Uniformization of Riemann Surfaces. Revisiting a hundred-year-old theorem, Heritage of European Mathematics, European Mathematical Society, 2016 (translated from the 2010 French original by Robert G. Burns) | MR | DOI | Zbl
[ST83] Manifolds with canonical coordinate charts: Some examples, Enseign. Math. (2), Volume 29 (1983), pp. 15-25 | MR | Zbl
[Thu97] Three-dimensional geometry and topology. Vol. 1., Princeton Mathematical Series, 35, Princeton University Press, 1997 | DOI | MR | Zbl
[Tra20] Gluing Delaunay ends to minimal -noids using the DPW method, Math. Ann., Volume 377 (2020) no. 3-4, pp. 1481-1508 | MR | DOI | Zbl
[Wit10] Mirror symmetry, Hitchin’s equations, and Langlands duality, The many facets of geometry. A tribute to Nigel Hitchin, Oxford University Press, 2010, pp. 113-128 | DOI | Zbl
[Wol89] The Teichmüller theory of harmonic maps, J. Differ. Geom., Volume 29 (1989) no. 2, pp. 449-479 | MR | DOI | Zbl