AdS manifolds with particles and earthquakes on singular surfaces.

*(English)*Zbl 1178.32009Let \(\Sigma\) be a closed surface with a hyperbolic metric \(m,\) and let \(\lambda\in{\mathcal{ML}}_\Sigma\) be a measured lamination on \(\Sigma\). Let \(\text{Teich}(\Sigma)\) denote the Teichmüller space of \(\Sigma\). W. P. Thurston [Symp. Warwick and Durham 1984, Lond. Math. Soc. Lect. Note Ser. 112, 91–112 (1986; Zbl 0628.57009)] defined the right earthquake \(E^r_\lambda:\text{Teich}(\Sigma)\to\text{Teich}(\Sigma)\) along \(\lambda\). The image \(E^r_\lambda(m)\) of \(m\) under the right earthquake along \(\lambda\) is obtained by cutting the surface \(\Sigma\) long the geodesics (in the support) of \(\lambda\), doing a fractional Dehn twist by the length corresponding to the weight associated to the curve by \(\lambda\), and then gluing back.

This defines a map \(E^r:{\mathcal{ML}}_\Sigma\times \text{Teich}(\Sigma)\to \text{Teich}(\Sigma)\).

Thurston proved that for any \(m\in \text{Teich}(\Sigma),\) the map \(E^r(m):{\mathcal{ML}}_\Sigma\to \text{Teich}(\Sigma)\) is bijective. In other words, for all \(m,m'\in \text{Teich}(\Sigma)\), there is a unique \(\lambda\in {\mathcal{ML}}_\Sigma\) such that \(E^r(m)=m'\).

A different proof of this was given by S. P. Kerchhoff [Ann. Math. (2) 117, 235–265 (1983; Zbl 0528.57008)]. G. Mess [Geom. Dedicata 126, 3–45 (2007; Zbl 1206.83117)] also discovered the earthquake theorem as a by-product of the geometric properties of globally hyperbolic maximal compact (GHMC) Anti-de Sitter (AdS) manifolds.

The authors of the present article consider a closed surface \(\Sigma\) with \(n\) distinct marked points \(x_1,\dots,x_n.\) They consider the hyperbolic metrics on \(\Sigma\) with cone singularities at \(x_i\) with angle \(\theta_i\) (\(i\in \overline{1,n}\) ). By a result of M. Troyanov [Trans. Am. Math. Soc. 324, No. 2, 793–821 (1991; Zbl 0724.53023)] and R. C. McOwen [Proc. Am. Math. Soc. 103, No. 1, 222–224 (1988; Zbl 0657.30033)] (see also [D. Hulin and M. Troyanov, Math. Ann. 293, No. 2, 277–315 (1992; Zbl 0799.53047) and R. C. McOwen, J. Math. Anal. Appl. 177, No. 1, 287–298 (1993; Zbl 0806.53040)]) given by \(\theta_i\), those metrics are in \(1-1\) correspondence with the conformal structures on \(\Sigma\), and thus, considered up to isotopies fixing \(x_i\), those metrics are parametrized by \(\text{Teich}_n(\Sigma)\), the Teichmüller space of \(\Sigma\) with \(n\) marked points.

The authors prove the earthquake theorem for hyperbolic surfaces with cone singularities where the total angle is less than \(\pi\). More precisely, they prove that any such two metrics are connected by a unique left earthquake. Secondly, they parametrize the space of globally hyperbolic AdS manifolds with particles-cone singularities (of given angle) along time-like lines by the Teichmüller space with some marked points (corresponding to the cone singularities).

This defines a map \(E^r:{\mathcal{ML}}_\Sigma\times \text{Teich}(\Sigma)\to \text{Teich}(\Sigma)\).

Thurston proved that for any \(m\in \text{Teich}(\Sigma),\) the map \(E^r(m):{\mathcal{ML}}_\Sigma\to \text{Teich}(\Sigma)\) is bijective. In other words, for all \(m,m'\in \text{Teich}(\Sigma)\), there is a unique \(\lambda\in {\mathcal{ML}}_\Sigma\) such that \(E^r(m)=m'\).

A different proof of this was given by S. P. Kerchhoff [Ann. Math. (2) 117, 235–265 (1983; Zbl 0528.57008)]. G. Mess [Geom. Dedicata 126, 3–45 (2007; Zbl 1206.83117)] also discovered the earthquake theorem as a by-product of the geometric properties of globally hyperbolic maximal compact (GHMC) Anti-de Sitter (AdS) manifolds.

The authors of the present article consider a closed surface \(\Sigma\) with \(n\) distinct marked points \(x_1,\dots,x_n.\) They consider the hyperbolic metrics on \(\Sigma\) with cone singularities at \(x_i\) with angle \(\theta_i\) (\(i\in \overline{1,n}\) ). By a result of M. Troyanov [Trans. Am. Math. Soc. 324, No. 2, 793–821 (1991; Zbl 0724.53023)] and R. C. McOwen [Proc. Am. Math. Soc. 103, No. 1, 222–224 (1988; Zbl 0657.30033)] (see also [D. Hulin and M. Troyanov, Math. Ann. 293, No. 2, 277–315 (1992; Zbl 0799.53047) and R. C. McOwen, J. Math. Anal. Appl. 177, No. 1, 287–298 (1993; Zbl 0806.53040)]) given by \(\theta_i\), those metrics are in \(1-1\) correspondence with the conformal structures on \(\Sigma\), and thus, considered up to isotopies fixing \(x_i\), those metrics are parametrized by \(\text{Teich}_n(\Sigma)\), the Teichmüller space of \(\Sigma\) with \(n\) marked points.

The authors prove the earthquake theorem for hyperbolic surfaces with cone singularities where the total angle is less than \(\pi\). More precisely, they prove that any such two metrics are connected by a unique left earthquake. Secondly, they parametrize the space of globally hyperbolic AdS manifolds with particles-cone singularities (of given angle) along time-like lines by the Teichmüller space with some marked points (corresponding to the cone singularities).

Reviewer: Yaşar Sözen (Istanbul)

##### MSC:

32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |

30F10 | Compact Riemann surfaces and uniformization |

30F45 | Conformal metrics (hyperbolic, Poincaré, distance functions) |

30F60 | Teichmüller theory for Riemann surfaces |