Conformal welding for critical Liouville quantum gravity

Abstract

Consider two critical Liouville quantum gravity surfaces (i.e., $\gamma$-LQG for $\gamma =2$), each with the topology of $\mathbb{H}$ and with infinite boundary length. We prove that there a.s. exists a conformal welding of the two surfaces, when the boundaries are identified according to quantum boundary length. This results in a critical LQG surface decorated by an independent SLE${}_4$. Combined with the proof of uniqueness for such a welding, recently established by McEnteggart, Miller, and Qian (2018), this shows that the welding operation is well-defined. Our result is a critical analogue of Sheffield's quantum gravity zipper theorem (2016), which shows that a similar conformal welding for subcritical LQG (i.e., $\gamma$-LQG for $\gamma \in (0,2)$) is well-defined.

27 pages, 3 figures. Minor changes

References in corpus (4)

• Critical Liouville measure as a limit of subcritical measures
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• Liouville quantum gravity weighted by conformal loop ensemble nesting statistics
• Regularity of the SLE${}_4$ uniformizing map and the SLE${}_8$ trace
• An elementary approach to quantum length of SLE
• On the $\log$-regularity of SLE${}_4$