UNIQUENESS of the WELDING PROBLEM for SLE and LIOUVILLE QUANTUM GRAVITY

Abstract

We give a simple set of geometric conditions on curves, in from to so that if is a homeomorphism which is conformal off with then is a conformal automorphism of. 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 is a non-space-filling curve in from to, and is a homeomorphism which is conformal on, and, are equal in distribution, then is a conformal automorphism of. Applying this result for establishes that the welding operation for critical LQG is well defined. Applying it for gives a new proof that the welding of two independent -stable looptrees of quantum disks to produce an on top of an independent -LQG surface is well defined.