Hofer's metrics and boundary depth

[Métriques de Hofer et profondeur de bord]

Usher, Michael

Annales scientifiques de l'École Normale Supérieure, Tome 46 (2013), p. 57-129 / Harvested from Numdam

Résumé

Nous montrons que si $(M,\omega )$ est une variété symplectique fermée qui admet un champ vectoriel hamiltonien non-trivial dont toutes les orbites fermées contractiles sont constantes, la métrique de Hofer sur le groupe des difféomorphismes hamiltoniens de $(M,\omega )$ a alors un diamètre infini et admet donc des espaces vectoriels normés plongés quasi-isométriquement et de dimension infinie. Une conclusion semblable s’applique à la métrique de Hofer sur différents espaces de sous-variétés lagrangiennes, y compris les sous-variétés hamiltoniennes isotopiques à la diagonale en $M\times M$ où $M$ satisfait à la condition dynamique ci-dessus. Pour prouver cela, nous utilisons les propriétés d’une quantité Floer-théorique appelée profondeur de bord, qui mesure la non-trivialité de l’opérateur limite sur le complexe de Floer de manière à encoder des informations robustes de topologie symplectique.

Abstract

We show that if $(M,\omega )$ is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer’s metric on the group of Hamiltonian diffeomorphisms of $(M,\omega )$ has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer’s metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in $M\times M$ when $M$ satisfies the above dynamical condition. To prove this, we use the properties of a Floer-theoretic quantity called the boundary depth, which measures the nontriviality of the boundary operator on the Floer complex in a way that encodes robust symplectic-topological information.

Détail

Publié le : 2013-01-01

Zbl 1271.53076

DOI : https://doi.org/10.24033/asens.2185
Classification:  53D22,  53D40
Mots clés: métrique de Hofer, difféomorphisme hamiltonien, sous-variété lagrangienne, complexe de Floer

Bibliographie

[1] W. Ballmann & P. Eberlein, Fundamental groups of manifolds of nonpositive curvature, J. Differential Geom. 25 (1987), 1-22. | MR 873453

[2] J.-F. Barraud & O. Cornea, Lagrangian intersections and the Serre spectral sequence, Ann. of Math. 166 (2007), 657-722. | MR 2373371

[3] A. Bertram & M. Thaddeus, On the quantum cohomology of a symmetric product of an algebraic curve, Duke Math. J. 108 (2001), 329-362. | MR 1833394

[4] P. Biran & O. Cornea, Quantum structures for Lagrangian submanifolds, preprint arXiv:0708.4221.

[5] P. Biran, L. Polterovich & D. Salamon, Propagation in Hamiltonian dynamics and relative symplectic homology, Duke Math. J. 119 (2003), 65-118. | MR 1991647

[6] Y. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J. 95 (1998), 213-226. | MR 1646550

[7] Y. V. Chekanov, Invariant Finsler metrics on the space of Lagrangian embeddings, Math. Z. 234 (2000), 605-619. | MR 1774099

[8] O. Cornea & F. Lalonde, Cluster homology, preprint arXiv:math/0508345.

[9] M. Entov & L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 2003 (2003), 1635-1676. | MR 1979584

[10] A. Floer, Morse theory for Lagrangian intersections, J. Differential Geom. 28 (1988), 513-547. | MR 965228

[11] A. Floer, Witten's complex and infinite-dimensional Morse theory, J. Differential Geom. 30 (1989), 207-221. | MR 1001276

[12] A. Floer, H. Hofer & D. Salamon, Transversality in elliptic Morse theory for the symplectic action, Duke Math. J. 80 (1995), 251-292. | MR 1360618

[13] K. Fukaya, Y.-G. Oh, H. Ohta & K. Ono, Canonical models of filtered $A_\infty $-algebras and Morse complexes, in New perspectives and challenges in symplectic field theory, CRM Proc. Lecture Notes 49, Amer. Math. Soc., 2009, 201-227. | MR 2555938

[14] K. Fukaya, Y.-G. Oh, H. Ohta & K. Ono, Lagrangian intersection Floer theory: Anomaly and obstruction, Studies in Adv. Math., 2009.

[15] K. Fukaya, Y.-G. Oh, H. Ohta & K. Ono, Anti-symplectic involution and Floer cohomology, preprint arXiv:0912:2642.

[16] K. Fukaya, Y.-G. Oh, H. Ohta & K. Ono, Displacement of polydisks and Lagrangian Floer theory, preprint arXiv:1102.4267.

[17] K. Fukaya & K. Ono, Arnold conjecture and Gromov-Witten invariant, Topology 38 (1999), 933-1048. | MR 1688434

[18] L. Gerritzen & U. Güntzer, Über Restklassennormen auf affinoiden Algebren, Invent. Math. 3 (1967), 71-74. | MR 214814

[19] R. E. Gompf, A new construction of symplectic manifolds, Ann. of Math. 142 (1995), 527-595. | MR 1356781

[20] H. Hofer, On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 25-38. | MR 1059642

[21] H. Hofer & D. Salamon, Floer homology and Novikov rings, in The Floer memorial volume, Progr. Math. 133, Birkhäuser, 1995, 483-524. | MR 1362838

[22] H. Hofer & C. Viterbo, The Weinstein conjecture in the presence of holomorphic spheres, Comm. Pure Appl. Math. 45 (1992), 583-622. | MR 1162367

[23] S. Hu & F. Lalonde, A relative Seidel morphism and the Albers map, Trans. Amer. Math. Soc. 362 (2010), 1135-1168. | MR 2563724

[24] H. Iritani, Convergence of quantum cohomology by quantum Lefschetz, J. reine angew. Math. 610 (2007), 29-69. | MR 2359850

[25] T. Kato, Perturbation theory for linear operators, Grundl. der Math. Wiss. 132, Springer, 1976. | MR 407617

[26] M. Khanevsky, Hofer's metric on the space of diameters, J. Topol. Anal. 1 (2009), 407-416. | MR 2597651

[27] F. Lalonde & D. Mcduff, The geometry of symplectic energy, Ann. of Math. 141 (1995), 349-371. | MR 1324138

[28] F. Lalonde & D. Mcduff, Hofer’s $L^\infty $-geometry: energy and stability of Hamiltonian flows. I, II, Invent. Math. 122 (1995), 1-33, 35-69. | MR 1354953

[29] F. Lalonde & C. Pestieau, Stabilisation of symplectic inequalities and applications, in Northern California Symplectic Geometry Seminar, Amer. Math. Soc. Transl. Ser. 2 196, Amer. Math. Soc., 1999, 63-71. | MR 1736214

[30] F. Lalonde & L. Polterovich, Symplectic diffeomorphisms as isometries of Hofer's norm, Topology 36 (1997), 711-727. | MR 1422431

[31] R. Leclercq, Spectral invariants in Lagrangian Floer theory, J. Mod. Dyn. 2 (2008), 249-286. | MR 2383268

[32] W. Li, The $\mathbb {Z}$-graded symplectic Floer cohomology of monotone Lagrangian sub-manifolds, Algebr. Geom. Topol. 4 (2004), 647-684. | MR 2100676

[33] G. Liu & G. Tian, Floer homology and Arnold conjecture, J. Differential Geom. 49 (1998), 1-74. | MR 1642105

[34] G. Lu, An explicit isomorphism between Floer homology and quantum homology, Pacific J. Math. 213 (2004), 319-363. | MR 2036923

[35] G. Lu, Gromov-Witten invariants and pseudo symplectic capacities, Israel J. Math. 156 (2006), 1-63. | MR 2282367

[36] D. Mcduff, Loops in the Hamiltonian group: a survey, in Symplectic topology and measure preserving dynamical systems, Contemp. Math. 512, Amer. Math. Soc., 2010, 127-148. | MR 2605315

[37] D. Mcduff, Monodromy in Hamiltonian Floer theory, Comment. Math. Helv. 85 (2010), 95-133. | MR 2563682

[38] D. Mcduff & D. Salamon, $J$-holomorphic curves and symplectic topology, American Mathematical Society Colloquium Publications 52, Amer. Math. Soc., 2004. | MR 2045629

[39] D. Mcduff & J. Slimowitz, Hofer-Zehnder capacity and length minimizing Hamiltonian paths, Geom. Topol. 5 (2001), 799-830. | MR 1871405

[40] D. Milinković, Action spectrum and Hofer's distance between Lagrangian submanifolds, Differential Geom. Appl. 17 (2002), 69-81. | MR 1912179

[41] Y.-G. Oh, Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks. I, Comm. Pure Appl. Math. 46 (1993), 949-993. | MR 1223659

[42] Y.-G. Oh, Spectral invariants and the length minimizing property of Hamiltonian paths, Asian J. Math. 9 (2005), 1-18. | MR 2150687

[43] Y.-G. Oh, Floer mini-max theory, the Cerf diagram, and the spectral invariants, J. Korean Math. Soc. 46 (2009), 363-447; Erratum 47 (2010), 1329-1330. | MR 2494501

[44] Y.-G. Oh & K. Zhu, Floer trajectories with immersed nodes and scale-dependent gluing, J. Symplectic Geom. 9 (2011), 483-636. | MR 2900788

[45] Y. Ostrover, A comparison of Hofer's metrics on Hamiltonian diffeomorphisms and Lagrangian submanifolds, Commun. Contemp. Math. 5 (2003), 803-811. | MR 2017719

[46] T. Perutz, Lagrangian matching invariants for fibred four-manifolds. I, Geom. Topol. 11 (2007), 759-828. | MR 2302502

[47] S. Piunikhin, D. Salamon & M. Schwarz, Symplectic Floer-Donaldson theory and quantum cohomology, in Contact and symplectic geometry (Cambridge, 1994), Publ. Newton Inst. 8, Cambridge Univ. Press, 1996, 171-200. | MR 1432464

[48] L. Polterovich, Hofer's diameter and Lagrangian intersections, Int. Math. Res. Not. 1998 (1998), 217-223. | MR 1609620

[49] P. Py, Quelques plats pour la métrique de Hofer, J. reine angew. Math. 620 (2008), 185-193. | MR 2427980

[50] J. Robbin & D. Salamon, The Maslov index for paths, Topology 32 (1993), 827-844. | MR 1241874

[51] D. Salamon & E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992), 1303-1360. | MR 1181727

[52] F. Schlenk, Applications of Hofer's geometry to Hamiltonian dynamics, Comment. Math. Helv. 81 (2006), 105-121. | MR 2208800

[53] M. Schwarz, Morse homology, Progress in Math. 111, Birkhäuser, 1993. | MR 1239174

[54] M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000), 419-461. | MR 1755825

[55] P. Seidel, $\pi _1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997), 1046-1095. | MR 1487754

[56] M. Usher, Spectral numbers in Floer theories, Compos. Math. 144 (2008), 1581-1592. | MR 2474322

[57] M. Usher, Duality in filtered Floer-Novikov complexes, J. Topol. Anal. 2 (2010), 233-258. | MR 2652908

[58] M. Usher, The sharp energy-capacity inequality, Commun. Contemp. Math. 12 (2010), 457-473. | MR 2661273

[59] M. Usher, Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds, Israel J. Math. 184 (2011), 1-57. | MR 2823968

[60] M. Usher, Many closed symplectic manifolds have infinite Hofer-Zehnder capacity, Trans. Amer. Math. Soc. 364 (2012), 5913-5943. | MR 2946937

[61] M. Usher, Linking and the morse complex, preprint arXiv:1207:0889.

[62] C. Viterbo, Intersection de sous-variétés lagrangiennes, fonctionnelles d'action et indice des systèmes hamiltoniens, Bull. Soc. Math. France 115 (1987), 361-390. | MR 926533

[63] J. A. Yorke, Periods of periodic solutions and the Lipschitz constant, Proc. Amer. Math. Soc. 22 (1969), 509-512. | MR 245916