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

TL;DR

This paper demonstrates that in $oxed{ ext{Liouville quantum gravity}}$, the surface can be reconstructed either through conformal welding or as a metric space quotient, extending previous results to the entire subcritical regime $oxed{ ext{(0,2)}}$ using GFF techniques.

Contribution

It extends the equivalence of metric gluing and conformal welding from $oxed{ ext{special case } ext{(} ext{ extonehalf} ext{)}}$ to all $oxed{ ext{(0,2)}}$-Liouville quantum gravity surfaces, employing GFF methods.

Findings

Established estimates relating distances, areas, and boundary lengths in LQG.

Proved bi-Hölder continuity of the LQG metric at the boundary.

Extended the metric space quotient representation to the entire subcritical regime.

Abstract

We consider the $\gamma$-Liouville quantum gravity (LQG) model for $\gamma \in (0,2)$, formally described by $e^{\gamma h}$ 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 $\gamma = \sqrt{8/3}$ to the whole subcritical regime $\gamma \in (0,2)$. Since the proof for $\gamma = \sqrt{8/3}$ used estimates for Brownian surfaces, which are equivalent to $\gamma$-LQG surfaces only when $\gamma=\sqrt{8/3}$, we instead use GFF techniques to establish estimates relating distances, areas and boundary lengths, as well as bi-H\"{o}lder continuity of the LQG metric w.r.t. the Euclidean metric at the boundary, which may be of independent interest.