# Comparison moduli spaces of Riemann surfaces

## TL;DR

This paper introduces comparison moduli spaces of nested Riemann surfaces, illustrating their role in complex analysis and Teichmueller theory, and discusses open problems in the field.

## Contribution

It defines comparison moduli spaces and demonstrates their applications in geometric function theory and Teichmueller theory, connecting various phenomena in complex analysis.

## Key findings

- Comparison moduli spaces unify diverse phenomena in complex analysis.
- Examples from geometric function theory and Teichmueller theory are reviewed.
- Open problems in classical and modern function theory are listed.

## Abstract

We define a kind of moduli space of nested surfaces and mappings, which we call a comparison moduli space. We review examples of such spaces in geometric function theory and modern Teichmueller theory, and illustrate how a wide range of phenomena in complex analysis are captured by this notion of moduli space. The paper includes a list of open problems in classical and modern function theory and Teichmueller theory ranging from general theoretical questions to specific technical problems.