Abstract
Given a smooth spacelike surface ∑ of negative curvature in Anti-de Sitter space of dimension 3, invariant by a representation p: π1 (S) → PSL2ℝ x PSL2ℝ where S is a closed oriented surface of genus ≥ 2, a canonical construction associates to ∑ a diffeomorphism φ∑ of S. It turns out that φ∑ is a symplectomorphism for the area forms of the two hyperbolic metrics h and h' on S induced by the action of p on ℍ2 x ℍ2. Using an algebraic construction related to the flux homomorphism, we give a new proof of the fact that φ∑ is the composition of a Hamiltonian symplectomorphism of (S, h) and the unique minimal Lagrangian diffeomorphism from (S, h) to (S, h’).
References
[1] Augustin Banyaga. Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique. Comment. Math. Helv., 53(2):174-227, 1978.10.1007/BF02566074Search in Google Scholar
[2] Thierry Barbot. Lorentzian Kleinian Groups. To appear in Handbook of Group Actions (ed. L. Ji, A. Papadopoulos and S.-T. Yau), 2016.Search in Google Scholar
[3] Thierry Barbot, François Béguin, and Abdelghani Zeghib. Constant mean curvature foliations of globally hyperbolic spacetimes locally modelled on AdS3. Geom. Dedicata, 126:71-129, 2007.10.1007/s10711-005-6560-7Search in Google Scholar
[4] Francesco Bonsante and Jean-Marc Schlenker. Maximal surfaces and the universal Teichmüller space. Invent. Math., 182(2):279-333, 2010.10.1007/s00222-010-0263-xSearch in Google Scholar
[5] Francesco Bonsante and Andrea Seppi. Area-preserving diffeomorphisms of hyperbolic plane and K-surfaces in Anti-de Sitter space. Preprint, ArXiv: 1610.05701, 2016.Search in Google Scholar
[6] Francesco Bonsante and Andrea Seppi. Equivariant maps into Anti-de Sitter space and the symplectic geometry of H2 × H2. Trans. Amer. Math. Soc., 2017. DOI 10.1090/tran/7417.10.1090/tran/7417Search in Google Scholar
[7] Eugenio Calabi. On the group of automorphisms of a symplectic manifold. pages 1-26, 1970.10.1515/9781400869312-002Search in Google Scholar
[8] C. J. Earle and J. Eells. The diffeomorphism group of a compact Riemann surface. Bull. Amer. Math. Soc., 73:557-559, 1967.10.1090/S0002-9904-1967-11746-4Search in Google Scholar
[9] Kirill Krasnov and Jean-Marc Schlenker. Minimal surfaces and particles in 3-manifolds. Geom. Dedicata, 126:187-254, 2007.10.1007/s10711-007-9132-1Search in Google Scholar
[10] François Labourie. Surfaces convexes dans l’espace hyperbolique et CP1-structures. J. London Math. Soc. (2), 45(3):549-565, 1992.10.1112/jlms/s2-45.3.549Search in Google Scholar
[11] Dusa McDuff. A survey of the topological properties of symplectomorphism groups. In Topology, geometry and quantum field theory, volume 308 of London Math. Soc. Lecture Note Ser., pages 173-193. Cambridge Univ. Press, Cambridge, 2004.10.1017/CBO9780511526398.010Search in Google Scholar
[12] Geoffrey Mess. Lorentz spacetimes of constant curvature. Geom. Dedicata, 126:3-45, 2007.10.1007/s10711-007-9155-7Search in Google Scholar
[13] Dusa McDuff and Dietmar Salamon. Introduction to symplectic topology. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, second edition, 1998.Search in Google Scholar
[14] Leonid Polterovich. The geometry of the group of symplectic diffeomorphisms. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2001.Search in Google Scholar
[15] Richard M. Schoen. The role of harmonic mappings in rigidity and deformation problems. In Complex geometry (Osaka, 1990), volume 143 of Lecture Notes in Pure and Appl. Math., pages 179-200. Dekker, New York, 1993.Search in Google Scholar
[16] Andrea Seppi. Minimal discs in hyperbolic space bounded by a quasicircle at infinity. Comment. Math. Helv., 91(4):807- 839, 2016.10.4171/CMH/403Search in Google Scholar
[17] Andrea Seppi. Maximal surfaces in Anti-de Sitter space, width of convex hulls and quasiconformal extensions of quasisymmetric homeomorphisms. To appear in Journal of the European Mathematical Society, 2017.Search in Google Scholar
[18] Francisco Torralbo. Minimal Lagrangian immersions in RH2 × RH2. In Symposium on the Differential Geometry of Submanifolds, pages 217-219. [s.n.], [s.l.], 2007Search in Google Scholar
© 2018
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.
