A new modular characterization of the hyperbolic plane

Abstract

We develop a natural and geometric way to realize the hyperbolic plane as the moduli space of marked genus 1 Riemann surfaces. To do so, a metric is defined on the Teichm\"uller space of the torus, inspired by Thurston's Lipschitz metric for the case of hyperbolic surfaces. Based on extremal Lipschitz maps, the Teichm\"uller space of the torus with this new metric is shown to be isometric to the hyperbolic plane under the usual identification. This also gives a new way to recover the complex-analytic Teichm\"uller metric via metric geometry on the underlying surfaces. Along the way, we prove a few results about this metric analogous to Thurston's Lipschitz metric in the case of hyperbolic surfaces, and analogous to the Teichm\"uller metric.