Limits of an increasing sequence of Riemann surfaces

TL;DR

This paper investigates the limits of increasing sequences of Riemann surfaces, characterizing their structure based on exhaustion by biholomorphic subsets and boundary component assumptions.

Contribution

It provides a detailed description of the limiting behavior of Riemann surfaces formed by increasing sequences, under various boundary conditions.

Findings

Characterization of limits of increasing sequences of Riemann surfaces.

Conditions under which the limit surface can be described explicitly.

Insights into the boundary behavior of the exhaustion domains.

Abstract

Let $M$ be a Riemann surface which admits an exhaustion by open subsets $M_j$ each of which is biholomorphic to a fixed domain $\Omega \subset \mathbb{C}$. We describe $M$ in terms of $\Omega$ under various assumptions on the boundary components of $\Omega$.