# Holomorphic quadratic differentials in Teichm\"uller theory

## TL;DR

This survey explores the role of holomorphic quadratic differentials in Teichmüller theory, emphasizing their connection with geometric structures on surfaces and discussing results for non-compact surfaces with poles at punctures.

## Contribution

It provides a comprehensive overview of how holomorphic quadratic differentials are integrated into Teichmüller theory and summarizes recent results for non-compact surfaces.

## Key findings

- Holomorphic quadratic differentials are central to various geometric structures.
- The survey highlights the relationship between different structures and quadratic differentials.
- Results for non-compact surfaces with poles are summarized.

## Abstract

This expository survey describes how holomorphic quadratic differentials arise in several aspects of Teichm\"uller theory, highlighting their relation with various geometric structures on surfaces. The final section summarizes results for non-compact surfaces of finite type, when the quadratic differential has poles of finite order at the punctures.