Closed continuations of Riemann surfaces

Abstract

Any open Riemann surface $R_0$ of finite genus $g$ can be conformally embedded into a closed Riemann surface of the same genus, that is, $R_0$ is realized as a subdomain of a closed Riemann surface of genus $g$. We are concerned with the set $M(R_0)$ of such closed Riemann surfaces. We formulate the problem in the Teichm\"{u}ller space setting to investigate geometric properties of $M(R_{0})$. We show, among other things, that $M(R_{0})$ is a closed Lipschitz domain homeomorphic to a closed ball provided that $R_0$ is nonanalytically finite.