Space of circle patterns on tori and its symplectic form

TL;DR

This paper explores the geometric structure of circle patterns on tori, demonstrating a symplectic form and confirming a conjecture about their topological equivalence to Teichmüller space.

Contribution

It establishes a symplectic form on the space of circle patterns and proves the space's homeomorphism to the Teichmüller space of the torus.

Findings

We show the Weil-Petersson form is non-degenerate on circle patterns.

The space of circle patterns is homeomorphic to the Teichmüller space of the torus.

The embedding relates circle patterns to complex projective structures.

Abstract

We consider circle patterns on closed tori equipped with complex projective structures. There is an embedding of the space of circle patterns to the Teichm\"{u}ller space of a punctured surface. Via the embedding, the Weil-Petersson symplectic form is pulled back to the space of circle patterns. We investigate its non-degeneracy. On the other hand, we also complete a conjecture that the space of circle patterns is homeomorphic to the Teichm\"{u}ller space of the closed torus.