Projective structures, laminations and the Thurston coordinates. I (Q2848828)

A complex projective structure on a surface is an atlas with values in \({\mathbb C}P^1\) and transition functions in \(PSL(2,{\mathbb C})\). The moduli space \(P(S)\) of marked complex projective structures on a closed surface of genus \(g>1\) is diffeomorphic to \({\mathbb C}^{6g-6}\) and it has a natural map to Teichmüller space \(T(S)\) because a complex projective structure determines a conformal structure. The Schwarzian derivative of the developing map can be used to identify \(P(S)\) with the cotangent bundle of \(T(S)\). Another parametrisation of \(P(S)\) is due to Thurston and is called grafting: to \(Y\in T(S)\) and a measured geodesic lamination \(\lambda\in {\mathcal ML}(S)\) one obtains a new projective structure by thickening the geodesic realization of \(\lambda\) in \(Y\) by inserting conformal annuli of the height determined by the transverse measure. According to an unpublished result of Thurston, grafting yields a homeomorphism \(T(S)\times {\mathcal ML}(S)\to P(S)\). The inverse of the grafting map is given by associating to a projective surface its conformal structure and the bending lamination of (the quotient of) the boundary of the hyperbolic convex hull of \({\mathbb C}P^1\setminus\Omega\), where \(\Omega\) is the image of the developing map. More information on Thurston's theorem can be found in [\textit{D. Dumas}, in: Handbook of Teichmüller theory. Volume II. Papadopoulos, Athanase (ed.), Zürich: European Mathematical Society (EMS). IRMA Lectures in Mathematics and Theoretical Physics 13, 455--508 (2009; Zbl 1196.30039)] and \textit{Y. Kamishima} and \textit{S. P. Tan} [in: Aspects of low dimensional manifolds. Matsumoto, Y. (ed.) et al., Tokyo: Kinokuniya Company Ltd.. Adv. Stud. Pure Math. 20, 263--299 (1992; Zbl 0798.53030)].NEWLINENEWLINEThe paper under review gives a detailed proof that the map \(T(S)\times {\mathcal ML}(S)\to P(S)\) is continuous. The proof that this map is indeed a homeomorphism will be given in a forthcoming paper.