Loewner theory for quasiconformal extensions: old and new

TL;DR

This survey explores the use of Loewner theory to study quasiconformal extensions of univalent functions, connecting classical and modern methods within Teichmüller theory.

Contribution

It provides a comprehensive overview of how Loewner theory has been applied to quasiconformal extensions, highlighting both historical and recent developments.

Findings

Unified framework for quasiconformal extensions via Loewner theory

Connections between Teichmüller theory and Loewner approaches

Summary of classical and modern techniques in the field

Abstract

This survey article gives an account of quasiconformal extensions of univalent functions with its motivational background from Teichm\"uller theory and classical and modern approaches based on Loewner theory.