Symplectic Wick rotations between moduli spaces of 3-manifolds

Given a closed hyperbolic surface $S$, let $\cQF$ denote the space of quasifuchsian hyperbolic
metrics on $S\times\R$ and $\cGH_{-1}$ the space of maximal globally hyperbolic anti-de Sitter
metrics on $S\times\R$. We describe natural maps between (parts of) $\cQF$ and $\cGH_{-1}$,
called ``Wick rotations'', defined in terms of special surfaces (e.g. minimal/maximal surfaces,
CMC surfaces, pleated surfaces) and prove that these maps are at least $C^1$ smooth and symplectic
with respect to the canonical symplectic structures on both $\cQF$ and $\cGH_{-1}$. Similar results
involving the spaces of globally hyperbolic de Sitter and Minkowski metrics are also described.

These 3-dimensional results are shown to be equivalent to purely 2-dimensional ones. Namely, consider
the double harmonic map $\cH:T^*\cT\to\cTT$, sending a conformal structure $c$ and a holomorphic quadratic
differential $q$ on $S$ to the pair of hyperbolic metrics $(m_L,m_R)$ such that the harmonic maps isotopic
to the identity from $(S,c)$ to $(S,m_L)$ and to $(S,m_R)$ have, respectively, Hopf differentials equal
to $i q$ and $-i q$, and the double earthquake map $\cE:\cT\times\cML\to\cTT$, sending a hyperbolic metric
$m$ and a measured lamination $l$ on $S$ to the pair $(E_L(m,l), E_R(m,l))$, where $E_L$ and
$E_R$ denote the left and right earthquakes. We describe how such 2-dimensional double maps
are related to 3-dimensional Wick rotations and prove that they are also $C^1$ smooth and symplectic.