Conformal removability of non-simple Schramm-Loewner evolutions
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arXiv
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| Format: | Preprint |
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2023
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| author | Kavvadias, Konstantinos Miller, Jason Schoug, Lukas |
| author_facet | Kavvadias, Konstantinos Miller, Jason Schoug, Lukas |
| contents | We consider the Schramm-Loewner evolution (SLE$_κ$) for $κ\in (4,8)$, which is the regime that the curve is self-intersecting but not space-filling. We let ${\mathcal K}$ be the set of $κ\in (4,8)$ for which the adjacency graph of connected components of the complement of an SLE$_κ$ 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 that this set is non-empty. We show that the range of an SLE$_κ$ for $κ\in {\mathcal K}$ 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 SLE$_κ$ for $κ\in (4,8)$ and establish a precise upper bound on the measure that it assigns to any Borel set in terms of its diameter. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_10857 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Conformal removability of non-simple Schramm-Loewner evolutions Kavvadias, Konstantinos Miller, Jason Schoug, Lukas Probability Mathematical Physics Complex Variables We consider the Schramm-Loewner evolution (SLE$_κ$) for $κ\in (4,8)$, which is the regime that the curve is self-intersecting but not space-filling. We let ${\mathcal K}$ be the set of $κ\in (4,8)$ for which the adjacency graph of connected components of the complement of an SLE$_κ$ 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 that this set is non-empty. We show that the range of an SLE$_κ$ for $κ\in {\mathcal K}$ 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 SLE$_κ$ for $κ\in (4,8)$ and establish a precise upper bound on the measure that it assigns to any Borel set in terms of its diameter. |
| title | Conformal removability of non-simple Schramm-Loewner evolutions |
| topic | Probability Mathematical Physics Complex Variables |
| url | https://arxiv.org/abs/2302.10857 |