Minimal surfaces and symplectic structures of moduli spaces

Abstract

Given a closed surface \(S\) of genus at least 2, we compare the symplectic structure of Taubes’ moduli space of minimal hyperbolic germs with the Goldman symplectic structure on the character variety \({\fancyscript{X}}(S, { PSL}(2,{\mathbb {C}}))\) and the affine cotangent symplectic structure on the space of complex projective structures \({\fancyscript{CP}}(S)\) given by the Schwarzian parametrization. This is done in restriction to the moduli space of almost-Fuchsian structures by involving a notion of renormalized volume, used to relate the geometry of a minimal surface in a hyperbolic 3-manifold to the geometry of its ideal conformal boundary.