Symplectic forms on the space of circle patterns

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 Teichmüller 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 Teichmüller space of the closed surface.