Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces

Fillastre, François; Seppi, Andrea

Annales Henri Lebesgue, Volume 3 (2020), pp. 873-899.

A well-known theorem of Wolpert shows that the Weil–Petersson symplectic form on Teichmüller space, computed on two infinitesimal twists along simple closed geodesics on a fixed hyperbolic surface, equals the sum of the cosines of the intersection angles. We define an infinitesimal deformation starting from a more general object, namely a balanced geodesic graph, by which any tangent vector to Teichmüller space can be represented. We then prove a generalization of Wolpert’s formula for these deformations. In the case of simple closed curves, we recover the theorem of Wolpert.

Un célèbre résultat de Wolpert dit que si on calcule la forme symplectique de Weil–Petersson sur l’espace de Teichmüller de deux twists infinitésimaux le long de géodésiques fermées simples sur une surface hyperbolique donnée, on obtient la somme des cosinus des angles d’intersection. Nous définissons des déformations infinitésimales à partir d’objets plus généraux qui sont des graphes géodésiques pondérés. Ils peuvent représenter n’importe quel vecteur tangent de l’espace de Teichmüller. Nous démontrons une formule qui généralise la formule de Wolpert à ces nouveaux objets. Dans le cas de courbes fermées simples, nous retrouvons exactement le résultat de Wolpert.

Classification: 32G15, 53C15, 53D30
Keywords: weighted geodesic cellulations, hyperbolic surfaces, Weil–Petersson form, Wolpert formula

[Bar05] Barbot, Thierry Globally hyperbolic flat space-times, J. Geom. Phys., Volume 53 (2005) no. 2, pp. 123-165 | DOI | MR | Zbl

[Ber60] Bers, Lipman Simultaneous uniformization, Bull. Am. Math. Soc., Volume 66 (1960), pp. 94-97 | DOI | MR | Zbl

[Bon86] Bonahon, Francis Bouts des variétés hyperboliques de dimension $3$, Ann. of Math., Volume 124 (1986) no. 1, pp. 71-158 | DOI | MR | Zbl

[Bon92] Bonahon, Francis Earthquakes on Riemann surfaces and on measured geodesic laminations, Trans. Am. Math. Soc., Volume 330 (1992) no. 1, pp. 69-95 | DOI | MR | Zbl

[Bon05] Bonsante, Francesco Flat spacetimes with compact hyperbolic Cauchy surfaces, J. Differ. Geom., Volume 69 (2005) no. 3, pp. 441-521 | DOI | MR | Zbl

[Bro03] Brock, Jeffrey F. The Weil–Petersson metric and volumes of 3-dimensional hyperbolic convex cores, J. Am. Math. Soc., Volume 16 (2003) no. 3, pp. 495-535 | DOI | MR | Zbl

[BS12] Bonsante, Francesco; Schlenker, Jean-Marc Fixed points of compositions of earthquakes, Duke Math. J., Volume 161 (2012) no. 6, pp. 1011-1054 | DOI | MR | Zbl

[BS16] Bonsante, Francesco; Seppi, Andrea On Codazzi tensors on a hyperbolic surface and flat Lorentzian geometry, Int. Math. Res. Not. IMRN (2016) no. 2, pp. 343-417 | DOI | MR | Zbl

[BS18] Bonsante, Francesco; Seppi, Andrea Area-preserving diffeomorphisms of the hyperbolic plane and $K$-surfaces in anti-de Sitter space, J. Topol., Volume 11 (2018) no. 2, pp. 420-468 | DOI | MR | Zbl

[CdV91] Colin de Verdière, Yves Comment rendre géodésique une triangulation d’une surface ?, Enseign. Math., Volume 37 (1991) no. 3-4, pp. 201-212 | MR | Zbl

[FS12] Fillastre, François; Schlenker, Jean-Marc Flippable tilings of constant curvature surfaces, Ill. J. Math., Volume 56 (2012) no. 4, pp. 1213-1256 | DOI | MR | Zbl

[FV16] Fillastre, François; Veronelli, Giona Lorentzian area measures and the Christoffel problem, Ann. Sc. Norm. Super. Pisa Cl. Sci., Volume 16 (2016) no. 2, pp. 383-467 | MR | Zbl

[Gol80] Goldman, William M. Discontinuous Groups and the Euler Class, Ph. D. Thesis, University of California, USA (1980) | MR

[Gol84] Goldman, William M. The symplectic nature of fundamental groups of surfaces, Adv. Math., Volume 54 (1984) no. 2, pp. 200-225 | DOI | MR | Zbl

[Koe09] Koebe, Paul Über die Uniformisierung der algebraischen Kurven. I, Math. Ann., Volume 67 (1909) no. 2, pp. 145-224 | DOI | MR | Zbl

[Lou15] Loustau, Brice The complex symplectic geometry of the deformation space of complex projective structures, Geom. Topol., Volume 19 (2015) no. 3, pp. 1737-1775 | DOI | MR | Zbl

[McM98] McMullen, Curtis Tracy Complex earthquakes and Teichmüller theory, J. Am. Math. Soc., Volume 11 (1998) no. 2, pp. 283-320 | DOI | MR | Zbl

[Mes07] Mess, Geoffrey Lorentz spacetimes of constant curvature, Geom. Dedicata, Volume 126 (2007), pp. 3-45 | DOI | MR | Zbl

[Rat06] Ratcliffe, John G. Foundations of hyperbolic manifolds, Graduate texts in mathematics, 149, Springer, 2006 | MR | Zbl

[SB01] Sözen, Yaşar; Bonahon, Francis The Weil–Petersson and Thurston symplectic forms, Duke Math. J., Volume 108 (2001) no. 3, pp. 581-597 | DOI | MR | Zbl

[Sep16] Seppi, Andrea Minimal discs in hyperbolic space bounded by a quasicircle at infinity, Comment. Math. Helv., Volume 91 (2016) no. 4, pp. 807-839 | DOI | MR | Zbl

[Tau04] Taubes, Clifford Henry Minimal surfaces in germs of hyperbolic 3-manifolds, Proceedings of the Casson Fest (Geometry and Topology Monographs), Volume 7, Geometry and Topology Publications (2004), pp. 69-100 | DOI | MR | Zbl

[Thu86] Thurston, William P. Earthquakes in two-dimensional hyperbolic geometry, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) (London Mathematical Society Lecture Note Series), Volume 112, Cambridge University Press, 1986, pp. 91-112 | MR | Zbl

[Uhl83] Uhlenbeck, Karen Keskulla Closed minimal surfaces in hyperbolic $3$-manifolds, Seminar on minimal submanifolds (Annals of Mathematics Studies), Volume 103, Princeton University Press, 1983, pp. 147-168 | MR | Zbl

[Wol81] Wolpert, Scott A. An elementary formula for the Fenchel–Nielsen twist, Comment. Math. Helv., Volume 56 (1981) no. 1, pp. 132-135 | DOI | MR | Zbl

[Wol83] Wolpert, Scott A. On the symplectic geometry of deformations of a hyperbolic surface, Ann. Math., Volume 117 (1983), pp. 207-234 | DOI | MR | Zbl