## Journal of Differential Geometry

### Generalized shear coordinates on the moduli spaces of three-dimensional spacetimes

#### Abstract

We introduce coordinates on the moduli spaces of maximal globally hyperbolic constant curvature 3d spacetimes with cusped Cauchy surfaces $S$. They are derived from the parametrization of the moduli spaces by the bundle of measured geodesic laminations over Teichmüller space of $S$ and can be viewed as analytic continuations of the shear coordinates on Teichmüller space. In terms of these coordinates, the gravitational symplectic structure takes a particularly simple form, which resembles the Weil–Petersson symplectic structure in shear coordinates, and is closely related to the cotangent bundle of Teichmüller space. We then consider the mapping class group action on the moduli spaces and show that it preserves the gravitational symplectic structure. This defines three distinct mapping class group actions on the cotangent bundle of Teichmüller space, corresponding to different values of the curvature.

#### Article information

Source
J. Differential Geom., Volume 103, Number 3 (2016), 425-474.

Dates
First available in Project Euclid: 14 July 2016

https://projecteuclid.org/euclid.jdg/1468517501

Digital Object Identifier
doi:10.4310/jdg/1468517501

Mathematical Reviews number (MathSciNet)
MR3523528

Zentralblatt MATH identifier
1347.83029

#### Citation

Meusburger, Catherine; Scarinci, Carlos. Generalized shear coordinates on the moduli spaces of three-dimensional spacetimes. J. Differential Geom. 103 (2016), no. 3, 425--474. doi:10.4310/jdg/1468517501. https://projecteuclid.org/euclid.jdg/1468517501