CaTherine wheels from trees and Liouville quantum gravity

TL;DR

This paper characterizes CaTherine wheels, a special class of space-filling curves, and demonstrates their unique correspondence with the geodesic tree in Liouville quantum gravity for all  .

Contribution

It provides necessary and sufficient conditions for topological trees to be realized as CaTherine wheels and constructs the unique such wheel for LQG geodesic trees.

Findings

Characterization of topological trees as CaTherine wheels.

Existence and uniqueness of the CaTherine wheel for LQG geodesic trees.

Construction of the space-filling curve exploring the LQG geodesic tree.

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 $\gamma$-Liouville quantum gravity (LQG) metric, for $\gamma \in (0,2)$. In other words, we construct the space-filling curve which is the contour exploration of the LQG geodesic tree.