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

Understand

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.