Pullback of symplectic forms to the space of circle patterns

TL;DR

This paper demonstrates that two fundamental symplectic forms, Goldman's and Weil-Petersson, coincide when pulled back to the space of circle patterns on surfaces, supporting the conjecture of a symplectic structure there.

Contribution

It proves the equality of two symplectic forms on the space of circle patterns and establishes smoothness, advancing the understanding of the geometric structure of this space.

Findings

The pullbacks of Goldman's and Weil-Petersson symplectic forms coincide.

The deformation space of circle patterns is smooth.

Supports the conjecture that this space has a natural symplectic structure.

Abstract

We consider circle patterns on surfaces with complex projective structures. We investigate two symplectic forms pulled back to the deformation space of circle patterns. The first one is Goldman's symplectic form on the space of complex projective structures on closed surfaces. The other is the Weil-Petersson symplectic form on the Teichm\"uller space of punctured surfaces. We show that their pullbacks to the space of circle patterns coincide. It is applied to prove the smoothness of the deformation space, which is an essential step to the conjecture that the space of circle patterns is homeomorphic to the Teichm\"uller space of the closed surface. We further conjecture that the pullback of the symplectic forms is non-degenerate and defines a symplectic structure on the space of circle patterns.