A new modular characterization of the hyperbolic plane

Mark Greenfield; Lizhen Ji

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arXiv · 1707.00818

A new modular characterization of the hyperbolic plane

Mark Greenfield·Lizhen Ji

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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 Teichmüller space of the torus, inspired by Thurston's Lipschitz metric for the case of hyperbolic surfaces. Based on extremal Lipschitz maps, the Teichmüller 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 Teichmüller 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 Teichmüller metric.

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BibTeX RIS

Mark Greenfield, Lizhen Ji. 2017-07-04. A new modular characterization of the hyperbolic plane. https://arxiv.org/abs/1707.00818

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