A symplectic map between hyperbolic and complex Teichmüller theory

Details A symplectic map between hyperbolic and complex Teichmüller theory A symplectic map between hyperbolic and complex Teichmüller theory

2008-05-30

arxiv.org/abs/0806.0010v2

Duke Math.J.150:331-356,2009

Mathematics

Differential Geometry

v2: clarified smoothness issues

Scientific paper

10.1215/00127094-2009-054

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.

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