%0 Journal Article
%T A symplectic map between hyperbolic and complex Teichm¨šller theory
%A Kirill Krasnov
%A Jean-Marc Schlenker
%J Mathematics
%D 2008
%I arXiv
%R 10.1215/00127094-2009-054
%X 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.
%U http://arxiv.org/abs/0806.0010v2