Hydrodynamic continuations of an open riemann surface of finite genus

Abstract

Every open Riemann surface Rof finite genus can be continued to a closed Riemann surface of the same genus. This classical result is usually proved by a local argument: one considers only a planar neighborhood of the ideal boundary of Rand applies the generalized uniformization theorem of Koebe. In the present paper we prove a continuation theorem of a global character: Let there be given a meromorphic function f on R with a special boundary behavior. Im(df shall be a distinguished harmonic differential of Ahlfors. Then there exists a closed Riemann surface R∗ of the same genus as Rand a meromorphic extension f ∗ of f onto R ∗ such that (i) R ∗\Rhas a vanishing area, (ii) f ∗ is holomorphic on R ∗\R, and (iii) Im f ∗ assumes a constant value on each boundary component of R with respect to R ∗ Since ƒ describes a hydrodynamic phenomenon on R, we call R ∗ a hydrodynamic continuation of R with respect to f. The ideal boundary Ris then realized on R ∗as a set of arcs on the streamlines of f with a total vanis...