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

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 $\varphi$ 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 $\Gamma \leftrightarrow \varphi$ in this case, giving a description of $\Gamma$ in terms of $\varphi$.

12 pages, 3 figures