Uniqueness of the welding problem for SLE and Liouville quantum gravity

Abstract

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

22 pages and 7 figures

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Cited by in corpus (6)

• Mating of trees for random planar maps and Liouville quantum gravity: a survey
• Conformal welding for critical Liouville quantum gravity
• Conformal welding of quantum disks
• On the $\log$-regularity of SLE${}_4$
• The bulk one-arm exponent for the CLE${}_{{\kappa }^{\prime }}$ percolations
• Regularity of the SLE${}_4$ uniformizing map and the SLE${}_8$ trace