CaTherine wheels from trees and Liouville quantum gravity
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866913044931543040 |
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| author | Calegari, Danny Gwynne, Ewain |
| author_facet | Calegari, Danny Gwynne, Ewain |
| contents | A CaTherine wheel is a space-filling curve $f : S^1\to S^2$ such that for every closed interval $J\subset S^1$, $f(J)$ is homeomorphic to a closed disk and $f(\partial J)$ is contained in $\partial f(J)$. A CaTherine wheel gives rise to a pair of disjoint, dense topological trees in $S^2$ which roughly speaking lie to the left and right of $f$. We give necessary and sufficient conditions for a topological tree in $S^2$ to arise as one of these trees for some CaTherine wheel $f$. We apply this result to show that there is a unique CaTherine wheel corresponding to the geodesic tree rooted at $\infty$ for the $γ$-Liouville quantum gravity (LQG) metric, for $γ\in (0,2)$. In other words, we construct the space-filling curve which is the contour exploration of the LQG geodesic tree. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_17170 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | CaTherine wheels from trees and Liouville quantum gravity Calegari, Danny Gwynne, Ewain Probability General Relativity and Quantum Cosmology Geometric Topology Metric Geometry A CaTherine wheel is a space-filling curve $f : S^1\to S^2$ such that for every closed interval $J\subset S^1$, $f(J)$ is homeomorphic to a closed disk and $f(\partial J)$ is contained in $\partial f(J)$. A CaTherine wheel gives rise to a pair of disjoint, dense topological trees in $S^2$ which roughly speaking lie to the left and right of $f$. We give necessary and sufficient conditions for a topological tree in $S^2$ to arise as one of these trees for some CaTherine wheel $f$. We apply this result to show that there is a unique CaTherine wheel corresponding to the geodesic tree rooted at $\infty$ for the $γ$-Liouville quantum gravity (LQG) metric, for $γ\in (0,2)$. In other words, we construct the space-filling curve which is the contour exploration of the LQG geodesic tree. |
| title | CaTherine wheels from trees and Liouville quantum gravity |
| topic | Probability General Relativity and Quantum Cosmology Geometric Topology Metric Geometry |
| url | https://arxiv.org/abs/2604.17170 |