A symplectic map between hyperbolic and complex Teichmüller theory

Abstract

Let $S$ be a closed, orientable surface of genus at least $2$. The space $T_H\times ML$, where $T_H$ is the “hyperbolic” Teichmüller space of $S$ and $\mathrm{ML}$ is the space of measured geodesic laminations on $S$, is naturally a real symplectic manifold. The space $\mathrm{CP}$ of complex projective structures on $S$ is a complex symplectic manifold. A relation between these spaces is provided by Thurston's grafting map $\mathrm{Gr}$. We prove that this map, although not smooth, is symplectic. The proof uses a variant of the renormalized volume defined for hyperbolic ends