Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Authors: Calegari, Danny, Gwynne, Ewain
Institution: University of Chicago
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7155035772
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