Quadratic differentials and foliations

Abstract

This paper concerns the interplay between the complex of a Riemann surface and the essentially Euclidean geometry induced by a quadratic differential. One aspect of this geometry is the trajectory structure of a quadratic differential which has long played a central role in Teichmfiller theory starting with Teichmiiller's proof of the existence and uniqueness of extremal maps. Ahlfors and Bers later gave proofs of that result. In other contexts, Jenkins and Strebel have studied quadratic differentials with closed trajectories. Starting from the dynamical problem of studying diffeomorphisms on a C ~ surface M, Thurston [17] invented measured ]ol~t io~. These are foliations with certain kinds of singularities and an invariantly defined transverse measure. A precise definition is given in Chapter I, w 1. This notion turns out to be the correct abstraction of the trajectory and metric induced by a quadratic differential. In this language our main statement says that given any measured ]oliation F on M and any complex X on M, there is a unique quadratic diHerential on the Riemann surface X whose horizontal trajectory realizes F. In particular any trajectory on one Riemann surface occurs uniquely on every Riemann surface of that genus. In the special case when the foliation has closed leaves, an analogous theorem was proved by Strebel [15]. Earlier Jenkins [13] had proved that quadratic differentials with closed trajectories existed as solutions of certain extremal problems. We deduce Strebel's theorem from ours in Chapter I, w 3. By identifying the space of measured foliations with the quadratic forms on a fixed Riemann surface, we are able to give an analytic and entirely different proof of a result of Thurston's [17]; that the space of projective classes of measured foliations is homeomorphic to a sphere. This is also done in Chapter I, w 3.