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Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds. (English) Zbl 1253.53085

Summary: We assign to each nondegenerate Hamiltonian on a closed symplectic manifold a Floer-theoretic quantity called its “boundary depth”, and establish basic results about how the boundary depths of different Hamiltonians are related. As applications, we prove that certain Hamiltonian symplectomorphisms supported in displaceable subsets have infinitely many nontrivial geometrically distinct periodic points, and we also significantly expand the class of co-isotropic submanifolds which are known to have positive displacement energy. For instance, any co-isotropic submanifold of contact type (in the sense of Bolle) in any closed symplectic manifold has positive displacement energy, as does any stable co-isotropic submanifold of a Stein manifold. We also show that any stable co-isotropic submanifold admits a Riemannian metric that makes its characteristic foliation totally geodesic, and that this latter, weaker, condition is enough to imply positive displacement energy under certain topological hypotheses.

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
57R58 Floer homology
37J10 Symplectic mappings, fixed points (dynamical systems) (MSC2010)
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[1] P. Albers and U. Frauenfelder, Leaf-wise intersections and Rabinowitz Floer homology, Journal of Topology and Analysis 2 (2010), 77–98. · Zbl 1196.53050 · doi:10.1142/S1793525310000276
[2] A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Commentarii Mathematici Helvetici 53 (1978), 174–227. · Zbl 0393.58007 · doi:10.1007/BF02566074
[3] P. Bolle, A contact condition for p-codimensional submanifolds of a symplectic manifold (2 p n), Mathematische Zeitschrift 227 (1998), 211–230. · Zbl 0894.53035 · doi:10.1007/PL00004373
[4] Yu. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Mathematical Journal 95 (1998), 213–226. · Zbl 0977.53077 · doi:10.1215/S0012-7094-98-09506-0
[5] O. Cornea and A. Ranicki, Rigidity and glueing for Morse and Novikov complexes, Journal of the European Mathematical Society 5 (2003), 343–394. · Zbl 1052.57052 · doi:10.1007/s10097-003-0052-6
[6] D. Dragnev, Symplectic rigidity, symplectic fixed points and global perturbations of Hamiltonian systems, Communications in Pure and Applied Mathematics 61 (2008), 346–370. · Zbl 1141.37022 · doi:10.1002/cpa.20203
[7] Y. Eliashberg, S.-S. Kim and L. Polterovich, Geometry of contact transformations and domains: orderability versus squeezing, Geometry and Topology 10 (2006), 1635–1747. · Zbl 1134.53044 · doi:10.2140/gt.2006.10.1635
[8] M. Entov and L. Polterovich, Rigid subsets of symplectic manifolds, Compositio Mathematica 145 (2009), 773–826. · Zbl 1230.53080 · doi:10.1112/S0010437X0900400X
[9] A. Floer, Symplectic fixed points and holomorphic spheres, Communications in Mathematical Physics 120 (1989), 575–611. · Zbl 0755.58022 · doi:10.1007/BF01260388
[10] K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariants, Topology 38 (1999), 933–1048. · Zbl 0946.53047 · doi:10.1016/S0040-9383(98)00042-1
[11] K. Fukaya, Y.-G. Oh, H. Ohta and K. Ono,Lagrangian intersection Floer theory – anomaly and obstruction, Preprint, 2000. · Zbl 1181.53003
[12] U. Frauenfelder and F. Schlenk, Hamiltonian dynamics on convex symplectic manifolds, Israel Journal of Mathematics 159 (2007), 1–56. · Zbl 1126.53056 · doi:10.1007/s11856-007-0037-3
[13] V. Ginzburg, Coisotropic intersections, Duke Mathematical Journal 140 (2007), 111–163. · Zbl 1129.53062 · doi:10.1215/S0012-7094-07-14014-6
[14] H. Gluck, Dynamical behavior of geodesic fields, Lecture Notes in Mathematics 819, Springer, Berlin, 1980, pp. 190–215. · Zbl 0448.58016
[15] M. Gotay, On coisotropic embeddings of presymplectic manifolds, Proceedings of the American Mathematical Society 84 (1982), 111–114. · Zbl 0476.53020 · doi:10.1090/S0002-9939-1982-0633290-X
[16] B. Gürel, Totally non-coisotropic displacement and its applications to Hamiltonian dynamics, Communications in Contemporary Mathematics 10 (2008), 1103–1128. · Zbl 1161.53077 · doi:10.1142/S0219199708003198
[17] H. Hofer and D. Salamon, Floer homology and Novikov rings, in The Floer Memorial Volume, Progress in Mathematics 133, Birkhauser, Basel, 1995, pp. 483–524. · Zbl 0842.58029
[18] D. Johnson and L. Whitt, Totally geodesic foliations, Journal of Differential Geometry 15 (1980), 225–235. · Zbl 0444.57017
[19] E. Kerman, Hofer’s geometry and Floer theory under the quantum limit, International Mathematics Research Notices 2008 (2008), article ID rnm137, 36 pages. · Zbl 1147.53070
[20] E. Kerman, Displacement energy of coisotropic submanifolds and Hofer’s geometry, The Journal of Modern Dynamics 2 (2008), 471–497. · Zbl 1149.53321
[21] E. Kerman and F. Lalonde, Length minimizing Hamiltonian paths for symplectically aspherical manifolds, Annales de l’Institut Fourier 53 (2003), 1503–1526. · Zbl 1113.53056 · doi:10.5802/aif.1986
[22] F. Laudenbach and J.-C. Sikorav, Hamiltonian disjunction and limits of Lagrangian submanifolds, International Mathematics Research Notices 1994 (1994), 8 pages. · Zbl 0812.53031 · doi:10.1155/S1073792894000176
[23] P. Lisca and G. Matić, Tight contact structures and Seiberg-Witten invariants, Inventiones Mathematicae 129 (1997), 509–525. · Zbl 0882.57008 · doi:10.1007/s002220050171
[24] G. Liu and G. Tian, Floer homology and Arnold conjecture, Journal of Differential Geometry 49 (1998), 1–74. · Zbl 0917.58009
[25] C.-M. Marle, Sous-variétés de rang constant d’une variété symplectique, in Third Schnepfenried Geometry Conference, Vol. 1 (Schnepfenried, 1982), Asterisque, 107–108 Soc. Math. France, Paris, 1983, pp. 69–86.
[26] J. Moser, A fixed point theorem in symplectic geometry, Acta Mathematica 141 (1978), 17–34. · Zbl 0382.53035 · doi:10.1007/BF02545741
[27] Y.-G. Oh, Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds, in The Breadth of Symplectic and Poisson Geometry, Progress inMathematics 232, Birkhäuser, Boston, 2005, pp. 525–570. · Zbl 1084.53076
[28] Y.-G. Oh, Spectral invariants and the length-minimizing property of Hamiltonian paths, The Asian Journal of Mathematics 9 (2005), 1–18. · Zbl 1084.53073 · doi:10.4310/AJM.2005.v9.n1.a1
[29] Y.-G. Oh, Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group, Duke Mathematical Journal 130 (2005), 199–295. · Zbl 1113.53054
[30] Y.-G. Oh, Lectures on Floer theory and spectral invariants of Hamiltonian flows, in Morse-Theoretic Methods in Nonlinear Analysis and in Symplectic Topology, NATO Science Series II: Mathematics, Physics and Chemistry, 217, Springer, Dordrecht, 2006, pp. 321–416. · Zbl 1089.53065
[31] Y.-G. Oh, Floer mini-max theory, the Cerf diagram, and the spectral invariants, Journal of the Korean Mathematical Society 46 (2009), 363–447. · Zbl 1180.53084 · doi:10.4134/JKMS.2009.46.2.363
[32] S. Piunikhin, D. Salamon and M. Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology, in Publ. Newton. Inst. (C. B. Thomas, ed.) 8, Cambridge University Press, Cambridge, 1996, pp. 1710–200. · Zbl 0874.53031
[33] L. Polterovich, An obstacle to non-Lagrangian intersections, in The Floer Memorial Volume, Progress in Mathematics 133, Birkhäuser, Basel, 1995, pp. 575–586. · Zbl 0847.58038
[34] D. Salamon, Lectures on Floer homology, in Symplectic Geometry and Topology (Park City, Utah, 1997), American Mathematical Society, Providence, RI, 1999. · Zbl 1031.53118
[35] D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Communications in Pure and Applied Mathematics 45 (1992), 1303–1360. · Zbl 0766.58023 · doi:10.1002/cpa.3160451004
[36] F. Schlenk, Applications of Hofer’s geometry to Hamiltonian dynamics, Commentarii Mathematici Helvetici 81 (2006), 105–121. · Zbl 1094.37031 · doi:10.4171/CMH/45
[37] M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific Journal of Mathematics 193 (2000), 419–461. · Zbl 1023.57020 · doi:10.2140/pjm.2000.193.419
[38] J.-C. Sikorav, Some properties of holomorphic curves in almost complex manifolds, in Holomorphic Curves in Symplectic Geometry, Progress in Mathematics 117, Birkhäuser, Basel, 1993, pp. 165–189.
[39] D. Sullivan, A foliation by geodesics is characterized by having no tangent homologies, The Journal of Pure and Applied Algebra 13 (1978), 101–104. · Zbl 0402.57015 · doi:10.1016/0022-4049(78)90046-4
[40] B. Tonnelier, A new condition of stability for coisotropic submanifolds, in preparation.
[41] M. Usher, Spectral numbers in Floer theories, Compositio Mathematica 144 (2008), 1581–1592. · Zbl 1151.53074 · doi:10.1112/S0010437X08003564
[42] M. Usher, Floer homology in disc bundles and symplectically twisted geodesic flows, The Journal of Modern Dynamics 3 (2009), 61–101. · Zbl 1186.53099 · doi:10.3934/jmd.2009.3.61
[43] M. Usher, The sharp energy-capacity inequality, Communications in Contemporary Mathematics 12 (2010), 457–473. · Zbl 1200.53077 · doi:10.1142/S0219199710003889
[44] J. Yorke, Periods of periodic solutions and the Lipschitz constant, Proceedings of the American Mathematical Society 22 (1969), 509–512. · Zbl 0184.12103 · doi:10.1090/S0002-9939-1969-0245916-7
[45] F. Ziltener, Coisotropic submanifolds, leafwise fixed points, and presymplectic embeddings, The Journal of Symplectic Geometry 8 (2010), 95–118. · Zbl 1208.57011 · doi:10.4310/JSG.2010.v8.n1.a6
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