Conformal removability of SLE$_4$

TL;DR

This paper proves that the range of SLE$_4$ curves is almost surely conformally removable, which has implications for the uniqueness of conformal weldings in Liouville quantum gravity.

Contribution

It establishes the conformal removability of SLE$_4$ curves and introduces a new sufficient condition for conformal removability applicable beyond simply connected domains.

Findings

SLE$_4$ curves are conformally removable almost surely.

The result confirms the uniqueness of conformal welding for critical Liouville quantum gravity surfaces.

A new criterion for conformal removability is developed.

Abstract

We consider the Schramm-Loewner evolution (SLE$_\kappa$) with $\kappa=4$, the critical value of $\kappa > 0$ at or below which SLE$_\kappa$ is a simple curve and above which it is self-intersecting. We show that the range of an SLE$_4$ curve is a.s. conformally removable, answering a question posed by Sheffield. Such curves arise as the conformal welding of a pair of independent critical ($\gamma=2$) Liouville quantum gravity (LQG) surfaces along their boundaries and our result implies that this conformal welding is unique. In order to establish this result, we give a new sufficient condition for a set $X \subseteq {\mathbf C}$ to be conformally removable which applies in the case that $X$ is not necessarily the boundary of a simply connected domain.