A symplectic map between hyperbolic and complex Teichm\"uller theory

TL;DR

This paper establishes a symplectic correspondence between hyperbolic and complex Teichmüller spaces via a map involving measured laminations and the Schwarzian derivative, using hyperbolic volume techniques.

Contribution

It proves that a natural map between the cotangent bundles of hyperbolic and complex Teichmüller spaces is symplectic, despite not being smooth.

Findings

The map is symplectic despite lack of smoothness.

Uses a variant of renormalized volume for hyperbolic ends.

Connects hyperbolic and complex structures through a geometric symplectic form.

Abstract

Let $S$ be a closed, orientable surface of genus at least 2. The cotangent bundle of the "hyperbolic'' Teichm\"uller space of $S$ can be identified with the space $\CP$ of complex projective structures on $S$ through measured laminations, while the cotangent bundle of the "complex'' Teichm\"uller space can be identified with $\CP$ through the Schwarzian derivative. We prove that the resulting map between the two cotangent spaces, although not smooth, is symplectic. The proof uses a variant of the renormalized volume defined for hyperbolic ends.