A projection from filling currents to Teichm\"uller space

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