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 Teichmüller 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.

73 pages

References in corpus (2)

• The holomorphic couch theorem
• Torus bundles and 2-forms on the universal family of Riemann surfaces