From curves to currents

Abstract

Many natural functionals on closed curves are known to extend continuously to the larger space of geodesic currents. For instance, the extension of hyperbolic length with respect to a fixed metric was a motivating example for the development of geodesic currents. We give a simple criterion on a curve functional that guarantees it extends to geodesic currents. It is easy to see that our criterion is satisfied for almost all the known examples of continuous functions on geodesic currents, while also applying to new examples like extremal length.