Abstract
In this paper, we develop a mini-max theory of the action functional over the semi-infinite cycles via the chain level Floer homology theory and construct spectral invariants of Hamiltonian paths on arbitrary, especially on nonexact and nonrational, compact symplectic manifold (M, ω). To each given time dependent Hamiltonian function H and quantum cohomology class 0 ≠ a ∈ QH*(M), we associate an invariant ρ(H; a) which varies continuously over H in the C 0-topology. This is obtained as the mini-max value over the semi-infinite cycles whose homology class is “dual” to the given quantum cohomology class a on the covering space \( \tilde \Omega _0 (M) \) of the contractible loop space Ω0(M). We call them the Novikov Floer cycles. We apply the spectral invariants to the study of Hamiltonian diffeomorphisms in sequels of this paper.
We assume that (M, ω) is strongly semipositive here, to be removed in a sequel to this paper.
This research was partially supported by the NSF grant DMS-9971446, by NSF grant DMS-9729992 at the Institute for Advanced Study, by a Vilas Associate Award at the University of Wisconsin, and by a grant of the Korean Young Scientist Prize.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Benci, V., and Rabinowitz, P., Critical point theorems for indefinite functionals, Invent. Math., 52 (1979), 241–273.
Berkovich, V., Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs, Vol. 33, American Mathematical Society, Providence, RI, 1990.
Chekanov, Y., Lagrangian intersections, symplectic energy and areas of holomorphic curves, Duke J. Math., 95 (1998), 213–226.
Entov, M., K-area, Hofer metric and geometry of conjugacy classes in Lie groups, Invent. Math., 146 (2001), 93–141.
Entov, M., Commutator length of symplectomorphisms, Comment. Math. Helv., 79-1 (2004), 58–104.
Entov, M., and Polterovich, L., Calabi quasimorphism and quantum homology, Internat. Math. Res. Notices, 30 (2003), 1635–1676.
Floer, A., Symplectic fixed points and holomorphic spheres, Comm. Math. Phys., 120 (1989), 575–611.
Fukaya, K., and Oh, Y.-G., Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math., 1 (1997), 96–180.
Fukaya, K., and Oh, Y.-G., in preparation.
Fukaya, K., Oh, Y.-G., Ohta, H., and Ono, K., Lagrangian Intersection Floer Theory: Anomaly and Obstruction, preprint, Kyoto University, Kyoto, 2000.
Fukaya, K., and Ono, K., Arnold conjecture and Gromov-Witten invariants, Topology, 38 (1999), 933–1048.
Gromov, M., Pseudo-holomorphic curves in symplectic manifolds, Invent. Math., 82 (1985), 307–347.
Guillemin, V., Lerman, E., and Sternberg, S., Symplectic Fibrations and Multiplicity Diagrams, Cambridge University Press, Cambridge, UK, 1996.
Harvey, F., and Lawson, B., Finite volume flows and Morse theory, Ann. Math., 153 (2001), 1–25.
Hofer, H., On the topological properties of symplectic maps, Proc. Roy. Soc. Edinburgh, 115 (1990), 25–38.
Hofer, H., and Salamon, D., Floer homology and Novikov rings, in Hofer, H., Taubes, C., Weinstein, A., and Zehnder, E., eds., The Floer Memorial Volume, Progress in Mathematics, Vol. 133, Birkhäuser, Basel, 1995, 483–524.
Lalonde, F., and McDuff, D., The geometry of symplectic energy, Ann. Math., 141 (1995), 349–371.
Liu, G., and Tian, G., Floer homology and Arnold’s conjecture, J. Differential Geom., 49 (1998), 1–74.
Liu, G., and Tian, G., On the equivalence of multiplicative structures in Floer homology and quantum homology, Acta Math. Sinica (English Ser.), 15-1 (1999), 53–80.
Lu, G., Arnold conjecture and PSS isomorphism between Floer homology and quantum homology, preprint, 2000.
Milinković, D., On equivalence of two constructions of invariants of Lagrangian submanifolds, Pacific J. Math., 195 (2000), 371–415.
Milinković, D., and Oh, Y.-G., Floer homology and stable Morse homology, J. Korean Math. Soc., 34 (1997), 1065–1087.
Milinković, D., and Oh, Y.-G., Generating functions versus the action functional: Stable Morse theory versus Floer theory, in Lalonde, F., ed., Geometry, Topology, and Dynamics: Proceedings of the Workshop on Geometry, Topology, and Dynamics Held at the CRM, Université de Montréal, June 26–30, 1995, CRM Proceedings and Lecture Notes, Vol. 15, American Mathematical Society, Providence, RI, 1998, 107–125.
Oh, Y.-G., Symplectic topology as the geometry of action functional I, J. Differential Geom., 46 (1997), 499–577.
Oh, Y.-G., Symplectic topology as the geometry of action functional II, Comm. Anal. Geom., 7 (1999), 1–55.
Oh, Y.-G., Gromov-Floer theory and disjunction energy of compact Lagrangian embeddings, Math. Rec. Lett., 4 (1997), 895–905.
Oh, Y.-G., Chain level Floer theory and Hofer’s geometry of the Hamiltonian diffeomorphism group, Asian J. Math., 6 (2002), 579–624; erratum, 7 (2003), 447–448.
Oh, Y.-G., Normalization of the Hamiltonian and the action spectrum, J. Korean Math. Soc., to appear; math.SG/0206090.
Oh, Y.-G., Mini-max theory, spectral invariants and geometry of the Hamiltonian diffeomorphism group, preprint, 2002; math.SG/0206092.
Oh, Y.-G., Spectral invariants and length minimizing property of Hamiltonian paths, Asian J. Math., to appear; math.SG/0212337.
Oh, Y.-G., Spectral invariants, analysis of the Floer moduli spaces and geometry of Hamiltonian diffeomorphisms, submitted.
Oh, Y.-G., Length minimizing property, Conley-Zehnder index and C 1-perturbation of Hamiltonian functions, submitted; math.SG/0402149.
Ostrover, Y., Acomparison of Hofer’s metrics on Hamiltonian diffeomorphisms and Lagrangian submanifolds, Comm. Contemp. Math., 5-5 (2003), 803–812.
Piunikhin, S., Salamon, D., and Schwarz, M., Symplectic Floer-Donaldson theory and quantum cohomology, in Thomas, C. B., ed., Contact and Symplectic Geometry, Publications of the Newton Institute, Vol. 8, Cambridge University Press, Cambridge, UK, 1996, 171–200.
Polterovich, L., Symplectic displacement energy for Lagrangian submanifolds, Ergodic Theory Dynam. Systems, 13 (1993), 357–367.
Polterovich, L., Gromov’s K-area and symplectic rigidity, Geom. Functional Anal., 6 (1996), 726–739.
Polterovich, L., The Geometry of the Group of Symplectic Diffeomorphisms, Lectures in Mathematics ETH Zürich, Birkhäuser, Basel, 2001.
Polterovich, L., private communication.
Rabinowitz, P., Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math., 31 (1978), 157–184.
Ruan, Y., Virtual neighborhood and pseudo-holomorphic curves, Turkish J. Math., 23 (1999), 161–231.
Schwarz, M., On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math., 193 (2000), 419–461.
Seidel, P., π 1 of symplectic diffeomorphism groups and invertibles in quantum homology rings, Geom. Functional Anal., 7 (1997), 1046–1095.
Viterbo, C., Symplectic topology as the geometry of generating functions, Math. Ann., 292 (1992), 685–710.
Weinstein, A., graduate course, University of California at Berkeley, Berkeley, CA, 1987.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Alan Weinstein in honor of his 60th birthday.
Rights and permissions
Copyright information
© 2005 Birkhäuser Boston
About this chapter
Cite this chapter
Oh, YG. (2005). Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds. In: Marsden, J.E., Ratiu, T.S. (eds) The Breadth of Symplectic and Poisson Geometry. Progress in Mathematics, vol 232. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4419-9_18
Download citation
DOI: https://doi.org/10.1007/0-8176-4419-9_18
Publisher Name: Birkhäuser Boston
Print ISBN: 978-0-8176-3565-7
Online ISBN: 978-0-8176-4419-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)