Closed continuations of Riemann surfaces

TL;DR

This paper studies how open Riemann surfaces of finite genus can be embedded into closed Riemann surfaces of the same genus, analyzing the geometric structure of the set of all such embeddings within Teichmüller space.

Contribution

It formulates the embedding problem in Teichmüller space and proves that the set of embeddings forms a closed Lipschitz domain homeomorphic to a closed ball for nonanalytically finite surfaces.

Findings

The set of embeddings is a closed Lipschitz domain.

This set is homeomorphic to a closed ball.

Results apply to nonanalytically finite 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.