Conformal removability of SLE${}_4$

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.

78 page, 7 figures