We give a simple set of geometric conditions on curves η, η in H from 0 to ∞ so that if ϕ : H → H is a homeomorphism which is conformal off η with ϕ(η) = η then ϕ is a conformal automorphism of 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 η is a non-space-filling SLEκ curve in H from 0 to ∞, and ϕ is a homeomorphism which is conformal on H \ η, and ϕ(η), η are equal in distribution, then ϕ is a conformal automorphism of H. Applying this result for κ = 4 establishes that the welding operation for critical (γ = 2) Liouville quantum gravity (LQG) is well-defined. Applying it for κ ∈ (4, 8) gives a new proof that the welding of two independent κ/4-stable looptrees of quantum disks to produce an SLEκ on top of an independent 4/ √ κ-LQG surface is well-defined.
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