Equivalence of metric gluing and conformal welding in $γ$-Liouville quantum gravity for $γ\in (0,2)$

Abstract

We consider the $γ$-Liouville quantum gravity (LQG) model for $γ\in (0,2)$, formally described by $e^{γh}$ where $h$ is a Gaussian free field on a planar domain $D$. Sheffield showed that when a certain type of LQG surface, called a quantum wedge, is decorated by an appropriate independent SLE curve, the wedge is cut into two independent surfaces which are themselves quantum wedges, and that the original surface can be recovered as a unique conformal welding. We prove that the original surface can also be obtained as a metric space quotient of the two wedges, extending results of Gwynne and Miller in the special case $γ= \sqrt{8/3}$ to the whole subcritical regime $γ\in (0,2)$. Since the proof for $γ= \sqrt{8/3}$ used estimates for Brownian surfaces, which are equivalent to $γ$-LQG surfaces only when $γ=\sqrt{8/3}$, we instead use GFF techniques to establish estimates relating distances, areas and boundary lengths, as well as bi-Hölder continuity of the LQG metric w.r.t. the Euclidean metric at the boundary, which may be of independent interest.

Figures (6)

• Figure 1: An illustration of Theorems \ref{['thm:main']} and \ref{['thm:16']} in the case of two thick wedges ($\mathfrak{w}_1$, $\mathfrak{w}_2 \ge \gamma^2/4$) which are glued along half their boundaries to yield a wedge of weight $\mathfrak{w} = \mathfrak{w}_1 + \mathfrak{w}_2$, then along the other half to yield a cone of weight $\mathfrak{w}$.
• Figure 2: An illustration of the arcs $K$, $K'$ and their neighbourhoods $U$, $U'$ from the proof of Prop. \ref{['prop:dint']}.
• Figure 3: The sets $\widetilde{U}$, $\widetilde{U}_n$ used in the proof of Prop. \ref{['prop:bihldrabi']}.
• Figure 4: We establish a bound on the narrowness of bottlenecks in $\mathrm{SLE}_\kappa$ curves for $\kappa \in (0,4)$. If $\mathrm{diam}\, \eta({[}s,t{]}) \gg |\eta(s)-\eta(t)|$, we have a large ball $B$ surrounded by the union of $\eta({[}s,t{]})$ and the line segment ${[}\eta(s),\eta(t){]}$. Since a Brownian motion started on $\varphi(\mathbb{R}+ir)$ is unlikely to hit $B$ before exiting $W^-$, a Brownian motion started on $\mathbb{R}+ir$ is unlikely to hit $\psi(B)$ before exiting $\mathbb{H}$, making $\mathrm{diam}\, \psi(B)$ small. This is impossible since the conformal coordinate change preserves quantum areas, which are bounded above and below by polynomials in Euclidean diameter, so the diameter of $\psi(B)$ cannot be smaller than a certain power of the diameter of $B$.
• Figure 5: We show a lower bound on the $\mu_h$-area near a boundary segment by using coupled $\mathrm{SLE}_{\gamma^2}(-1;-1)$ curves to cut a wedge into independent surfaces each of which have a positive chance of accumulating some positive amount of $\mu_h$-area within a small $\mathfrak{d}_h$-distance of the boundary.

Theorems & Definitions (62)

• Definition 1.1: metric gluing
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Proposition 1.7
• Proposition 1.8
• Proposition 1.9
• Lemma 3.1