Minimal surfaces and symplectic structures of moduli spaces. (English) Zbl 1323.53091
The author deals with a closed oriented surface \(S\) of genus larger than two, and with certain moduli spaces associated to \(S\). The central role is played by the deformation space of almost-Fuchsian structures \(\mathcal{A F}(S)\) (see [K. K. Uhlenbeck, Ann. Math. Stud. 103, 147–168 (1983; Zbl 0529.53007)]), since it can be viewed as an open subspace of either Taubes’ moduli space \(\mathcal{H},\) the deformation space \(\mathcal{C P}(S)\) of complex projective structures, or the character variety \(\mathcal{X}(S,\mathrm{PSL}_2(\mathbb{C}))\). The author compares the Goldman symplectic structure \(\omega_G\) on \(\mathcal{X}(S,\mathrm{PSL}_2(\mathbb{C}))\) restricted to \(\mathcal{A F}(S)\) with the symplectic structure \(\omega_{\mathcal H}\) of Taubes’ moduli space of minimal hyperbolic germs and the canonical cotangent symplectic structure \(\omega_{\text{can}}\) on the cotangent bundle to the Teichmüller space \(T^*\mathcal{T}(S)\). To this aim, the notion of renormalized volume for almost-Fuchsian manifolds is used; see [the author, Geom. Topol. 19, No. 3, 1737–1775 (2015; Zbl 1318.53097)].
Reviewer: Simona Druta-Romaniuc (Iaşi)
MSC:
| 53D30 | Symplectic structures of moduli spaces |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
Keywords:
minimal surfaces; symplectic structures; character varieties; Teichmüller theory; almost-Fuchsian structures; renormalized volumeReferences:
| [1] | Ahlfors, L.V.: Some remarks on Teichmüller’s space of Riemann surfaces. Ann. Math. 74, 171-191 (1961) · Zbl 0146.30602 |
| [2] | Atiyah, M.F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Philos. Trans. R. Soc. Lond. A 308(1505), 523-615 (1983) · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017 |
| [3] | Chen, B.-Y.: Riemannian submanifolds. In: Handbook of differential geometry, vol. I, pp. 187-418. North-Holland, Amsterdam (2000) · Zbl 0910.53036 |
| [4] | Dumas, D.: Complex projective structures. In: Handbook of Teichmüller theory. Vol. II, volume 13 of IRMA Lect. Math. Theor. Phys., pp. 455-508. Eur. Math. Soc., Zürich (2009) · Zbl 1196.30039 |
| [5] | Goldman, W.M.: The symplectic nature of fundamental groups of surfaces. Adv. Math. 54(2), 200-225 (1984) · Zbl 0574.32032 · doi:10.1016/0001-8708(84)90040-9 |
| [6] | Goldman, W.M.: The complex-symplectic geometry of SL \[(2, \mathbb{C}C)\]-characters over surfaces. In: Algebraic groups and arithmetic, pp. 375-407. Tata Inst. Fund. Res., Mumbai (2004) · Zbl 1089.53060 |
| [7] | Hejhal, D.A.: Monodromy groups and linearly polymorphic functions. Acta Math. 135(1), 1-55 (1975) · Zbl 0333.34002 · doi:10.1007/BF02392015 |
| [8] | Heusener, M., Porti, J.: The variety of characters in \[\text{ PSL }_2(\mathbb{C})\] PSL2(C). Bol. Soc. Mat. Mexi. 10(Special Issue), 221-237 (2004) · Zbl 1100.57014 |
| [9] | Hodgson, C.D., Rivin, I.: A characterization of compact convex polyhedra in hyperbolic \[33\]-space. Invent. Math. 111(1), 77-111 (1993) · Zbl 0784.52013 · doi:10.1007/BF01231281 |
| [10] | Hopf, H.: Über Flächen mit einer Relation zwischen den Hauptkrümmungen. Math. Nachr. 4, 232-249 (1951) · Zbl 0042.15703 · doi:10.1002/mana.3210040122 |
| [11] | Krasnov, K., Schlenker, J.-M.: Minimal surfaces and particles in 3-manifolds. Geom. Dedicata 126, 187-254 (2007) · Zbl 1126.53037 · doi:10.1007/s10711-007-9132-1 |
| [12] | Krasnov, K., Schlenker, J.-M.: On the renormalized volume of hyperbolic 3-manifolds. Commun. Math. Phys. 279(3), 637-668 (2008) · Zbl 1155.53036 · doi:10.1007/s00220-008-0423-7 |
| [13] | Krasnov, K., Schlenker, J.-M.: The Weil-Petersson metric and the renormalized volume of hyperbolic 3-manifolds. In: Handbook of Teichmüller theory. Volume III, volume 17 of IRMA Lect. Math. Theor. Phys., pp. 779-819. Eur. Math. Soc., Zürich (2012) · Zbl 1256.30001 |
| [14] | Loustau, B.: The symplectic geometry of the deformation space of complex projective structures over a surface. PhD thesis, University of Toulouse III, Toulouse (2011) |
| [15] | Loustau, B.: The symplectic geometry of the deformation space of complex projective structures on a surface. arXiv:1406.1821 (2014) · Zbl 0988.32012 |
| [16] | McMullen, C.T.: The moduli space of Riemann surfaces is Kähler hyperbolic. Ann. Math. 151(1), 327-357 (2000) · Zbl 0988.32012 · doi:10.2307/121120 |
| [17] | Rivin, I., Schlenker, J.-M.: The Schläfli formula in Einstein manifolds with boundary. Electron. Res. Announc. Am. Math. Soc. 5, 18-23 (1999). (electronic) · Zbl 0910.53036 · doi:10.1090/S1079-6762-99-00057-8 |
| [18] | Taubes, C.H.: Minimal surfaces in germs of hyperbolic 3-manifolds. In: Proceedings of the Casson Fest, volume 7 of Geom. Topol. Monogr., pp. 69-100 (electronic). Geom. Topol. Publ., Coventry (2004) · Zbl 1087.53011 |
| [19] | Thurston, W.P.: Three-dimensional geometry and topology. Vol. 1 volume 35 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ (1997). Edited by Silvio Levy. · Zbl 0873.57001 |
| [20] | Uhlenbeck, K.K.: Closed minimal surfaces in hyperbolic 3-manifolds. In: Seminar on minimal submanifolds, volume 103 of Annals of Mathematical Studies, pp. 147-168. Princeton University Press, Princeton, NJ (1983) · Zbl 0529.53007 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
