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arXiv · 0806.0010

A symplectic map between hyperbolic and complex Teichmüller theory

Kirill Krasnov·Jean-Marc Schlenker
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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.

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Kirill Krasnov, Jean-Marc Schlenker. 2009-01-20. A symplectic map between hyperbolic and complex Teichmüller theory. https://doi.org/10.1215/00127094-2009-054

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