A symplectic map between hyperbolic and complex Teichmüller theory
Kirill Krasnov and Jean-Marc Schlenker
Source: Duke Math. J. Volume 150, Number 2 (2009), 331-356.
Abstract
Let $S$ be a closed, orientable surface of genus at least $2$. The space ${\mathcal T}_H\times {\mathcal ML}$, where ${\mathcal T}_H$ is the “hyperbolic” Teichmüller space of $S$ and ${\mathcal ML}$ is the space of measured geodesic laminations on $S$, is naturally a real symplectic manifold. The space ${\mathcal CP}$ of complex projective structures on $S$ is a complex symplectic manifold. A relation between these spaces is provided by Thurston's grafting map ${\rm Gr}$. We prove that this map, although not smooth, is symplectic. The proof uses a variant of the renormalized volume defined for hyperbolic ends
Primary Subjects: 30F60
Secondary Subjects: 32G15
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Permanent link to this document: http://projecteuclid.org/euclid.dmj/1255699343
Digital Object Identifier: doi:10.1215/00127094-2009-054
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