Generalization of a formula of Wolpert for balanced geodesic graphs on closed hyperbolic surfaces
Annales Henri Lebesgue, Volume 3 (2020) , pp. 873-899.

Metadata

Keywordsweighted geodesic cellulations, hyperbolic surfaces, Weil–Petersson form, Wolpert formula

Abstract

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.


References

[Bar05] Barbot, Thierry Globally hyperbolic flat space-times, J. Geom. Phys., Volume 53 (2005) no. 2, pp. 123-165 | Article | MR 2110829 | Zbl 1087.53065

[Ber60] Bers, Lipman Simultaneous uniformization, Bull. Am. Math. Soc., Volume 66 (1960), pp. 94-97 | Article | MR 0111834 | Zbl 0090.05101

[Bon86] Bonahon, Francis Bouts des variétés hyperboliques de dimension 3, Ann. of Math., Volume 124 (1986) no. 1, pp. 71-158 | Article | MR 847953 | Zbl 0671.57008

[Bon92] Bonahon, Francis Earthquakes on Riemann surfaces and on measured geodesic laminations, Trans. Am. Math. Soc., Volume 330 (1992) no. 1, pp. 69-95 | Article | MR 1049611 | Zbl 0754.32010

[Bon05] Bonsante, Francesco Flat spacetimes with compact hyperbolic Cauchy surfaces, J. Differ. Geom., Volume 69 (2005) no. 3, pp. 441-521 | Article | MR 2170277 | Zbl 1094.53063

[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 | Article | MR 1969203 | Zbl 1059.30036

[BS12] Bonsante, Francesco; Schlenker, Jean-Marc Fixed points of compositions of earthquakes, Duke Math. J., Volume 161 (2012) no. 6, pp. 1011-1054 | Article | MR 2913100 | Zbl 1244.32007

[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 | Article | MR 3493421 | Zbl 1336.53034

[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 | Article | MR 3789829 | Zbl 1396.53094

[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 1151746 | Zbl 0753.57009

[FS12] Fillastre, François; Schlenker, Jean-Marc Flippable tilings of constant curvature surfaces, Ill. J. Math., Volume 56 (2012) no. 4, pp. 1213-1256 | Article | MR 3231480 | Zbl 1296.52011

[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 3559607 | Zbl 1358.52012

[Gol80] Goldman, William M. Discontinuous Groups and the Euler Class (1980) (Ph. D. Thesis) | MR 2630832

[Gol84] Goldman, William M. The symplectic nature of fundamental groups of surfaces, Adv. Math., Volume 54 (1984) no. 2, pp. 200-225 | Article | MR 762512 | Zbl 0574.32032

[Koe09] Koebe, Paul Über die Uniformisierung der algebraischen Kurven. I, Math. Ann., Volume 67 (1909) no. 2, pp. 145-224 | Article | MR 1511526 | Zbl 40.0470.01

[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 | Article | MR 3352248 | Zbl 1318.53097

[McM98] McMullen, Curtis Tracy Complex earthquakes and Teichmüller theory, J. Am. Math. Soc., Volume 11 (1998) no. 2, pp. 283-320 | Article | MR 1478844 | Zbl 0890.30031

[Mes07] Mess, Geoffrey Lorentz spacetimes of constant curvature, Geom. Dedicata, Volume 126 (2007), pp. 3-45 | Article | MR 2328921 | Zbl 1206.83117

[Rat06] Ratcliffe, John G. Foundations of hyperbolic manifolds, Graduate texts in mathematics, Volume 149, Springer, 2006 | MR 2249478 | Zbl 1106.51009

[SB01] Sözen, Yaşar; Bonahon, Francis The Weil–Petersson and Thurston symplectic forms, Duke Math. J., Volume 108 (2001) no. 3, pp. 581-597 | Article | MR 1838662 | Zbl 1014.32009

[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 | Article | MR 3566524 | Zbl 1356.53063

[Tau04] Taubes, Clifford Henry Minimal surfaces in germs of hyperbolic 3-manifolds, Proceedings of the Casson Fest (Geometry and Topology Monographs) Volume 7 (2004), pp. 69-100 | Article | MR 2172479 | Zbl 1087.53011

[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 903860 | Zbl 0628.57009

[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 795233 | Zbl 0529.53007

[Wol81] Wolpert, Scott A. An elementary formula for the Fenchel–Nielsen twist, Comment. Math. Helv., Volume 56 (1981) no. 1, pp. 132-135 | Article | MR 615620 | Zbl 0467.30036

[Wol83] Wolpert, Scott A. On the symplectic geometry of deformations of a hyperbolic surface, Ann. Math., Volume 117 (1983), pp. 207-234 | Article | MR 690844 | Zbl 0518.30040