Equivariant maps into Anti-de Sitter space and the symplectic geometry of $\mathbb H^2\times \mathbb H^2$
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- by Francesco Bonsante and Andrea Seppi PDF
- Trans. Amer. Math. Soc. 371 (2019), 5433-5459 Request permission
Abstract:
Given two Fuchsian representations $\rho _l$ and $\rho _r$ of the fundamental group of a closed oriented surface $S$ of genus $\geq 2$, we study the relation between Lagrangian submanifolds of $M_\rho =(\mathbb {H}^2/\rho _l(\pi _1(S)))\times (\mathbb {H}^2/\rho _r(\pi _1(S)))$ and $\rho$-equivariant embeddings $\sigma$ of $\widetilde S$ into Anti-de Sitter space, where $\rho =(\rho _l,\rho _r)$ is the corresponding representation into $\mathrm {PSL}_2\mathbb R\times \mathrm {PSL}_2\mathbb R$. It is known that, if $\sigma$ is a maximal embedding, then its Gauss map takes values in the unique minimal Lagrangian submanifold $\Lambda _{\mathrm {ML}}$ of $M_\rho$.
We show that, given any $\rho$-equivariant embedding $\sigma$, its Gauss map gives a Lagrangian submanifold Hamiltonian isotopic to $\Lambda _{\mathrm {ML}}$. Conversely, any Lagrangian submanifold Hamiltonian isotopic to $\Lambda _{\mathrm {ML}}$ is associated to some equivariant embedding into the future unit tangent bundle of the universal cover of Anti-de Sitter space.
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Additional Information
- Francesco Bonsante
- Affiliation: Dipartimento di Matematica “Felice Casorati”, Università degli Studi di Pavia, Via Ferrata 5, 27100, Pavia, Italy
- MR Author ID: 769339
- Email: bonfra07@unipv.it
- Andrea Seppi
- Affiliation: Dipartimento di Matematica “Felice Casorati”, Università degli Studi di Pavia, Via Ferrata 5, 27100, Pavia, Italy
- Address at time of publication: CNRS and Université Grenoble Alpes, 100 Rue des Mathématiques, 38610, Giéres, France
- MR Author ID: 1134288
- Email: andrea.seppi@univ-grenoble-alpes.fr
- Received by editor(s): May 31, 2017
- Received by editor(s) in revised form: September 27, 2017
- Published electronically: December 21, 2018
- Additional Notes: The authors were partially supported by FIRB 2010 project “Low dimensional geometry and topology” (RBFR10GHHH003) and by PRIN 2012 project “Moduli strutture algebriche e loro applicazioni”.
The authors are members of the national research group GNSAGA - © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 5433-5459
- MSC (2010): Primary 53C50; Secondary 57M50, 53D12
- DOI: https://doi.org/10.1090/tran/7417
- MathSciNet review: 3937298