A new modular characterization of the hyperbolic plane

TL;DR

This paper introduces a novel geometric approach to model the hyperbolic plane as the moduli space of marked genus 1 Riemann surfaces, using a new metric inspired by Thurston's Lipschitz metric.

Contribution

It defines a new metric on the Teichmüller space of the torus, showing it is isometric to the hyperbolic plane and providing a new perspective on classical metrics.

Findings

The new metric is isometric to the hyperbolic plane.

The approach recovers the classical Teichmüller metric via metric geometry.

Analogous properties to Thurston's Lipschitz metric are established.

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.