Conformal weldings in the Loewner equation and Weil--Petersson quasislit-disks

TL;DR

This paper explores the relationship between conformal weldings and Loewner equations in the context of Weil--Petersson quasislit-disks, providing a new description of slit growth in terms of conformal weldings.

Contribution

It offers a novel analysis of the conformal welding maps associated with Weil--Petersson quasislit-disks, extending understanding of Loewner evolutions and weldings.

Findings

Characterization of slit growth via conformal weldings

Extension of Loewner theory to Weil--Petersson quasislit-disks

New description of slit geometry in terms of welding maps

Abstract

A simple arc $\Gamma = \gamma(0, T]$, growing into the unit disk $\mathbb D$ from its boundary, generates a driving term $\xi$ and a conformal welding $\phi$ through the Loewner differential equation. When $\Gamma$ is the slit of a Weil--Petersson quasislit-disk $\mathbb D\setminus\Gamma$, the Loewner transform and its inverse $\Gamma \leftrightarrow \xi$ have been well understood due to Y. Wang's work. We investigate the maps $\Gamma \leftrightarrow \phi$ in this case, giving a description of $\Gamma$ in terms of $\phi$.