Reference : Symplectic Wick rotations between moduli spaces of 3-manifolds
 Document type : Scientific journals : Article Discipline(s) : Physical, chemical, mathematical & earth Sciences : Mathematics To cite this reference: http://hdl.handle.net/10993/19099
 Title : Symplectic Wick rotations between moduli spaces of 3-manifolds Language : English Author, co-author : scarinci, carlos [University of Nottingham > Mathematics] Schlenker, Jean-Marc [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] Publication date : 2014 Journal title : Annali della Scuola Normale Superiore di Pisa: Classe di Scienze Publisher : SNS Pisa Peer reviewed : Yes Audience : International ISSN : 0391-173x City : Pisa Country : Italy Abstract : [en] 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. Permalink : http://hdl.handle.net/10993/19099

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