2023/02/21 by Kavvadias, Konstantinos, Miller, Jason, Schoug, Lukas
#Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR)
paper · doi:10.48550/arxiv.2302.10857
We consider the Schramm-Loewner evolution (SLEκ) for κ∈ (4,8), which is the regime that the curve is self-intersecting but not space-filling. We let \mathcal K be the set of κ∈ (4,8) for which the adjacency graph of connected components of the complement of an SLEκ is a.s. connected, meaning that for every pair of complementary components U, V there exist complementary components U1,…,Un with U1 = U, Un = V, and ∂ Ui ∩ ∂ Ui+1 ≠ ∅ for each 1 ≤ i ≤ n-1. We show that the range of an SLEκ for κ∈ \mathcal 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 for κ∈ (4,8) and establish a precise upper bound on the measure that it assigns to any Borel set in terms of its diameter.