A surface \(S\) is said to be of finite type if: (1) it is of the form \(S=\hat{S}\setminus V\) where \((\hat{S}, V)\) is a closed oriented surface of genus \(g\geq 0\) with a set of \(r\geq 0\) marked points \(V=\{p_1,\dots,p_r\}\), (2) the fundamental group of \(S\) is non abelian, equivalently \(2-2g-r<0\). The aim of this survey is to describe, for every \(S\) of finite type, and for every \(k=0, \pm 1\), the geometry of 3-dimensional maximal globally hyperbolic Lorentzian space-times of constant curvature \(k\) that contain a complete Cauchy surface homeomorphic to \(S\).
For the entire collection see [Zbl 1158.30001].