Uniqueness of the Welding Problem for Sle and Liouville Quantum Gravity

Abstract

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.

The geodesics in Liouville quantum gravity are not Schramm–Loewner evolutions

“…The conformal removability of a path in LQG is important because it implies that a conformal welding in which the path arises as the gluing interface is uniquely determined (see, e.g., [9,26,38]). In the case γ = √ 8/3, the conformal removability of geodesics is especially important as it is shown in [28] that metric balls in the Brownian map can be decomposed into independent slices obtained by cutting the metric ball along the geodesics from its outer boundary to its center (see Fig.…”

Conformal welding for critical Liouville quantum gravity

“…In particular, for a class of homeomorphisms defined in terms subcritical LQG measures (γ -LQG for γ ∈ (0, 2)) existence and uniqueness of the conformal welding was established by Sheffield [28], and the interface η was proven to have the law of an SLE κ with κ = γ 2 ∈ (0, 4). Uniqueness of a random conformal welding where the interface η has the law of an SLE 4 was recently established by McEnteggart, Miller, and Qian [20].…”

“…Observe that the conformal welding in this corollary is not proven to be the unique conformal welding among all possible conformal weldings; since it is assumed in Theorem 1.3 that the curves ϕ(η) and η both have the law of SLE 4 curves, we only obtain uniqueness among the weldings for which the interface has this law. The uniqueness result can be strengthened to curves a.s. satisfying certain deterministic geometric properties by using the stronger variant of Theorem 1.3 found in [20,Theorem 2].…”

“…Such random surfaces give rise to random conformal welding problems, for instance, when the homeomorphism φ corresponds to gluing the boundaries of two discs according to their LQG-boundary lengths. Weldings of this type have been studied in several recent works [3,4,11,20,28]. In particular, for a class of homeomorphisms defined in terms subcritical LQG measures (γ -LQG for γ ∈ (0, 2)) existence and uniqueness of the conformal welding was established by Sheffield [28], and the interface η was proven to have the law of an SLE κ with κ = γ 2 ∈ (0, 4).…”

“…The following uniqueness result concerning the conformal welding problem of Theorem 1.2 was recently established in [20,Theorem 2].…”