A length comparison theorem for geodesic currents

TL;DR

This paper establishes a comparison between the geometry of geodesic currents and their associated length-minimizing metrics on thick components of a surface, providing new insights into their relationship.

Contribution

It proves that on thick components, the geometries of geodesic currents and their length-minimizing metrics are comparable up to a scalar, and characterizes these components using the length function.

Findings

Geometries of geodesic currents and metrics are comparable on thick components.

Thick components can be characterized solely by the length function of the current.

The comparison depends only on the current and the surface topology.

Abstract

We work with the space $\mathcal C(S)$ of geodesic currents on a closed surface $S$ of negative Euler characteristic. By prior work of the author with Sebastian Hensel, each filling geodesic current $\mu$ has a unique length-minimizing metric $X$ in Teichm\"uller space. In this paper, we show that, on so-called thick components of $X$, the geometries of $\mu$ and $X$ are comparable, up to a scalar depending only on $\mu$ and the topology of $S$. We also characterize thick components of the projection using only the length function of $\mu$.