On the Local Structure of the Trajectories of a Quadratic Differential

1. Teichmuller was the first to bring out the important role played by quadratic differentials in extremal problems in the theory of functions. He indicated [3] the behavior in the small of certain important curves associated with them. It remained for Schaeffer and Spencer [2, Chap. Ill] to carry out this study in full detail. While they speak explicitly only of hyperelliptic differentials, the study of the local structure is the same in the general case. The object of the present paper is to show how, by a somewhat different approach to the problem, a considerable simplification is obtained of the technical details involved in this discussion. In this local study we are concerned only with interior points of a Riemann surface. On an oriented Riemann surface tyt, open or closed, a quadratic differential is an entity which assigns to every local uniformizing parameter z of dt a function Qiz) meromorphic in the neighborhood associated with z and satisfying the following condition. If z* is a second local uniformizing parameter of dt whose neighborhood on dt overlaps that of z and Q*(z*) is the corresponding function associated with z*, then at common points of the neighborhoods of z and z* we have