Uniqueness of the welding problem for SLE and Liouville quantum gravity

TL;DR

This paper establishes geometric conditions ensuring conformal automorphisms from homeomorphisms conformal off certain curves, with applications to SLE and Liouville quantum gravity welding problems, confirming well-definedness in critical and supercritical cases.

Contribution

It provides new geometric criteria for conformal rigidity in the presence of curves, applied to random conformal welding in SLE and LQG contexts, including critical and supercritical regimes.

Findings

Conformal automorphism results under geometric conditions on curves.

Validation of welding operation for critical ($b3=2$) LQG.

New proof of welding for $ ext{SLE}_$ on quantum surfaces.

Abstract

We give a simple set of geometric conditions on curves $\eta$, $\tilde{\eta}$ in ${\mathbf H}$ from $0$ to $\infty$ so that if $\varphi \colon {\mathbf H} \to {\mathbf H}$ is a homeomorphism which is conformal off $\eta$ with $\varphi(\eta) = \tilde{\eta}$ then $\varphi$ is a conformal automorphism of ${\mathbf H}$. Our motivation comes from the fact that it is possible to apply our result to random conformal welding problems related to the Schramm-Loewner evolution (SLE) and Liouville quantum gravity (LQG). In particular, we show that if $\eta$ is a non-space-filling SLE$_\kappa$ curve in ${\mathbf H}$ from $0$ to $\infty$ and $\varphi$ is a homeomorphism which is conformal on ${\mathbf H} \setminus \eta$ and $\varphi(\eta)$, $\eta$ are equal in distribution then $\varphi$ is a conformal automorphism of ${\mathbf H}$. Applying this result for $\kappa=4$ establishes that the welding operation for critical ($\gamma=2$) Liouville quantum gravity (LQG) is well-defined. Applying it for $\kappa \in (4,8)$ gives a new proof that the welding of two independent $\kappa/4$-stable looptrees of quantum disks to produce an SLE$_\kappa$ on top of an independent $4/\sqrt{\kappa}$-LQG surface is well-defined.