Complex Projective Structures

In this chapter we discuss the theory of complex projective structures on compact surfaces and its connections with Teichmüller theory, 2-and 3-dimensional hyperbolic geometry, and representations of surface groups into PSL 2 (C). Roughly speaking, a complex projective structure is a type of 2-dimensional geometry in which Möbius transformations play the role of geometric congru-ences (this is made precise below). Such structures are abundant—hyperbolic, spherical, and Euclidean metrics on surfaces all provide examples of projective structures, since each of these constant-curvature 2-dimensional geometries has a model in which its isometries are Möbius maps. However, these examples are not representative of the general situation, since most projective structures are not induced by locally homogeneous Riemannian metrics. Developing a more accurate picture of a general projective structure is the goal of the first half of the chapter (§ §2–4). After some definitions and preliminary discussion (in §2), we present the complex-analytic theory of projective structures in §3. This theory has its roots in the study of automorphic functions and differential equations by Klein [67, Part 1], Poincaré [95], Riemann [100], and others in the late nineteenth century (see [48] [47, §1] for further historical discussion and references), while its more recent history is closely linked to developments in Teichmüller theory and deformations of Fuchsian and Kleinian groups (e. In this analytic approach, a projective structure is represented by a holo-morphic quadratic differential on a Riemann surface, which is extracted from the geometric data using a Möbius-invariant differential operator, the Schwarzian derivative. The inverse of this construction describes every projective structure in terms of holomorphic solutions to a linear ordinary differential equation (the Schwarzian equation). In this way, many properties of projective structures and their moduli can be established using tools from complex function theory. However, in spite of the success of these techniques, the analytic theory is somewhat detached from the underlying geometry. In particular, the analytic parameterization of projective structures does not involve an explicit geometric construction, such as one has in the description of hyperbolic surfaces by gluing polygons. In §4 we describe a more direct and geometric construction of complex projective structures using grafting, a gluing operation on surfaces which is also suggested by the work of the nineteenth-century geometers (e.g. [68]), but whose significance in complex projective geometry has only recently been fully appreciated. Grafting was used by Maskit [83], Hejhal [47], and Sullivan-Thurston [109] to construct certain …