Uniqueness of the welding problem for SLE and Liouville quantum gravity

We give a simple set of geometric conditions on curves $\eta$, $\tilde{\eta }$ in $\mathbf{H}$ from $0$ to $\mathrm{\infty }$ so that if $\varphi \colon {\mathbf H} \to {\mathbf H}$isahomeomorphismwhichisconformaloff$\eta$with$\varphi(\eta) = \tilde{\eta}$then$\varphi$isaconformalautomorphismof${\mathbf H}$. 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 $\mathrm{\infty }$ and $\phi$ is a homeomorphism which is conformal on $\mathbf{H}\setminus \eta$ and $\phi (\eta )$, $\eta$ are equal in distribution then $\phi$ 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 $\kappa \mathrm{/}4$-stable looptrees of quantum disks to produce an SLE${}_{\kappa }$ on top of an independent $4\mathrm{/}\sqrt{\kappa }$-LQG surface is well-defined.