Minimal surfaces and symplectic structures of moduli spaces

TL;DR

This paper explores the relationships between different symplectic structures on moduli spaces associated with a closed surface, using minimal surfaces, hyperbolic geometry, and complex projective structures.

Contribution

It establishes a comparison between the symplectic structure of Taubes' minimal hyperbolic germs and the Goldman and Schwarzian structures on related moduli spaces.

Findings

Identifies a correspondence between minimal surface geometry and symplectic structures.

Introduces a notion of renormalized volume linking hyperbolic 3-manifolds to conformal boundary geometry.

Provides a framework for understanding the interplay of minimal surfaces and complex structures in hyperbolic geometry.

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 X(S, PSL(2,C)) and the affine cotangent symplectic structure on the space of complex projective structures 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.