A symplectic map between hyperbolic and complex Teichmüller theory

Abstract

Let $S$ be a closed, orientable surface of genus at least 2. The cotangent bundle of the "hyperbolic'' Teichmüller 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'' Teichmüller 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.

v2: clarified smoothness issues

References in corpus (4)

• On the renormalized volume of hyperbolic 3-manifolds
• Minimal surfaces and particles in 3-manifolds
• Liouville action and Weil-Petersson metric on deformation spaces, global Kleinian reciprocity and holography
• Projective structures, grafting, and measured laminations

Cited by in corpus (5)

• No Ensemble Averaging Below the Black Hole Threshold
• The dual Bonahon-Schläfli formula
• The convex core of quasifuchsian manifolds with particles
• A quantization of moduli spaces of 3-dimensional gravity
• The infimum of the dual volume of convex co-compact hyperbolic $3$-manifolds