A projection from filling currents to Teichmüller space

Abstract

Let S S be a closed, genus g g surface. The space of geodesic currents on S S encompasses the set of closed curves up to homotopy, as well as Teichmüller space, and many other spaces of structures on S S . We show that one can define a mapping class group equivariant, length minimizing projection from the set of filling geodesic currents down to Teichmüller space, and prove some basic properties of this projection to show that it is well-behaved.