Conformal removability of non-simple Schramm-Loewner evolutions

Abstract We consider the Schramm-Loewner evolution ( ${\mathrm{SLE}}_{\kappa }$ SLE κ ) for $\kappa \in (4,8)$ κ ∈ ( 4 , 8 ) , which is the regime that the curve is self-intersecting but not space-filling. We let $\mathcal{K}$ K be the set of $\kappa \in (4,8)$ κ ∈ ( 4 , 8 ) for which the adjacency graph of connected components of the complement of an ${\mathrm{SLE}}_{\kappa }$ SLE κ is a.s. connected, meaning that for every pair of complementary components $U, V$ U , V there exist complementary components $U_{1},\ldots ,U_{n}$ U 1 , … , U n with $U_{1} = U$ U 1 = U , $U_{n} = V$ U n = V , and $\partial U_{i} \cap \partial U_{i+1} \neq \emptyset $ ∂ U i ∩ ∂ U i + 1 ≠ ∅ for each $1 \leq i \leq n-1$ 1 ≤ i ≤ n − 1 . It was proved by Gwynne and Pfeffer [11] that this set is non-empty. We show that the range of an ${\mathrm{SLE}}_{\kappa }$ SLE κ for $\kappa \in \mathcal{K}$ κ ∈ K 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 ${\mathrm{SLE}}_{\kappa }$ SLE κ for $\kappa \in (4,8)$