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

A simple arc $\mathrm{\Gamma }=\gamma (0,T]$, 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 $\mathrm{\Gamma }$ is the slit of a Weil–Petersson quasislit-disk $\mathbb{D}\setminus \mathrm{\Gamma }$, the Loewner transform and its inverse $\mathrm{\Gamma }\leftrightarrow \xi$ have been well understood due to Y. Wang’s work. We investigate the maps $\mathrm{\Gamma }\leftrightarrow \phi$ in this case, giving a description of $\mathrm{\Gamma }$ in terms of $\phi$.