A sewing theorem for quadratic differentials

Abstract

Quadratic differentials $$ \mathfrak{Q}(z)d{z^2} $$ on a finite Riemann surface with poles of order not exceeding two are considered. The existence of such a differential with prescribed metric characteristics is proved. These characteristics are the following: the leading coefficients in the expansions of the function $$ \mathfrak{Q}(z) $$ in neighborhoods of its poles of order two, the conformal modules of the ring domains, and the heights of the strip domains in the decomposition of the Riemann surface defined by the structure of trajectories of this differential. Bibliography: 5 titles.