Conformal removability of non-simple Schramm-Loewner evolutions
- 1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA
- 2Statistical Laboratory, Centre for Mathematical Sciences, University of Cambridge, Cambridge, CB3 0WB, UK
- 3Department of Mathematics, KTH Royal Institute of Technology, Stockholm, 100 44, Sweden
Abstract
We consider the Schramm-Loewner evolution ( SLEκ SLE κ ) for κ ∈ (4,8) κ ∈ ( 4 , 8 ) , which is the regime that the curve is self-intersecting but not space-filling. We let 𝒦 K be the set of κ ∈ (4,8) κ ∈ ( 4 , 8 ) for which the adjacency graph of connected components of the complement of an SLEκ SLE κ is a.s. connected, meaning that for every pair of complementary components $U, V$ U , V there exist complementary components U1,… ,Un U 1 , … , U n with U1 = U U 1 = U , Un = V U n = V , and $\partial U_{i} \cap \partial U_{i+1} \neq \emptyset $ ∂ U i ∩ ∂ U i + 1 ≠ ∅ for each 1 ≤ i ≤ 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 SLEκ SLE κ for κ ∈ 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 SLEκ SLE κ for κ ∈ (4,8)
Related
- Conformal non-removability of the Brownian graph
- A simple construction of smooth counterexamples to the Loewner and Carathéodory conjectures
- Conformal boundary rigidity for simple Finsler metrics
- Removability of non-isolated singularities for Einstein metrics and RCD spaces
- Every Closed Oriented Surface of Positive Genus Is Loewner
BibTeX
@article{kavvadias2026conformal,
title = {{Conformal removability of non-simple Schramm-Loewner evolutions}},
author = {Konstantinos Kavvadias and Jason Miller and Lukas Schoug},
journal = {Inventiones mathematicae},
year = {2026},
month = jun,
doi = {10.1007/s00222-026-01427-3},
url = {https://doi.org/10.1007/s00222-026-01427-3},
note = {CC BY},
}
RIS
TY - JOUR
TI - Conformal removability of non-simple Schramm-Loewner evolutions
AU - Konstantinos Kavvadias
AU - Jason Miller
AU - Lukas Schoug
PY - 2026
DA - 2026/06/12
JO - Inventiones mathematicae
DO - 10.1007/s00222-026-01427-3
UR - https://doi.org/10.1007/s00222-026-01427-3
AB - We consider the Schramm-Loewner evolution ( SLE_κ SLE κ ) for κ ∈ (4,8) κ ∈ ( 4 , 8 ) , which is the regime that the curve is self-intersecting but not space-filling. We let 𝒦 K be the set of κ ∈ (4,8) κ ∈ ( 4 , 8 ) for which the adjacency graph of connected components of the complement of an SLE_κ SLE κ is a.s. connected, meaning that for every pair of complementary components $U, V$ U , V there exist complementary components U_1,… ,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 ≤ i ≤ 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 SLE_κ SLE κ for κ ∈ 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 SLE_κ SLE κ for κ ∈ (4,8)
SN - 0020-9910
SN - 1432-1297
ER -