A projection from filling currents to Teichm\"uller space

TL;DR

This paper introduces a new projection from the space of filling geodesic currents to Teichmüller space for a closed surface, which is equivariant and length-minimizing, enhancing understanding of the geometric structures on the surface.

Contribution

It defines and analyzes a length-minimizing, mapping class group equivariant projection from filling geodesic currents to Teichmüller space, establishing its basic properties.

Findings

The projection is well-defined and length-minimizing.

It is equivariant under the mapping class group.

The projection has desirable geometric properties.

Abstract

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.