Conformal removability of non-simple Schramm-Loewner evolutions

Abstract

We consider the Schramm-Loewner evolution ($\SLE_\kappa$) for $\kappa \in (4,8)$, which is the regime that the curve is self-intersecting but not space-filling. We let $\adjcon$ be the set of $\kappa \in (4,8)$ for which the adjacency graph of connected components of the complement of an $\SLE_\kappa$ is a.s.\ connected, meaning that for every pair of complementary components $U, V$ there exist complementary components $U_1,\ldots,U_n$ with $U_1 = U$, $U_n = V$, and $\partial U_i \cap \partial U_{i+1} \neq \emptyset$ for each $1 \leq i \leq n-1$. It was proved by Gwynne and Pfeffer \cite{gp2020adj} that this set is non-empty. We show that the range of an $\SLE_\kappa$ for $\kappa \in \adjcon$ is a.s.\ conformally removable, which answers a question of Sheffield. As a step in the proof, we construct the canonical conformally covariant volume measure on the cut points of an $\SLE_\kappa$ for $\kappa \in (4,8)$ and establish a precise upper bound on the measure that it assigns to any Borel set in terms of its diameter.