Geometry and Topology Seminar: "From curves to currents"

Abstract: Geodesic currents are measures introduced by Bonahon in 1986 that realize a suitable closure of the space of closed curves on a surface. They are analogous to measured laminations for simple closed curves. Many geometric structures on surfaces, such as hyperbolic structures or half-translation structures can be realized as geodesic currents. Bonahon proved that the notion of hyperbolic length for curves extends to geodesic currents. Since then, many other functions defined on the space of curves have been proven to extend to currents, such as negatively curved lengths, lengths from singular flat structures or stable lengths for surface groups. In this talk, we explain how a function defined on the space of curves satisfying some simple conditions can be extended continuously to geodesic currents. The most important of these is that the function decreases under smoothing of essential crossings.

Our theorem subsumes previous extension results. Furthermore, it extends functions that had not been considered before, such as extremal length. This is joint work with Dylan Thurston.

Host: Michael Landry