Space of circle patterns on tori and its symplectic form

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 Teichmüller 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 Teichmüller space of the closed torus.

arXiv comment14 pages. arXiv admin note: text overlap with arXiv:2404.17458