Duke Mathematical Journal

Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group

Yong-Geun Oh

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this article, we apply spectral invariants constructed in [Oh5] and [6] to the study of Hamiltonian diffeomorphisms of closed symplectic manifolds ( M , ω ) . Using spectral invariants, we first construct an invariant norm called the spectral norm on the Hamiltonian diffeomorphism group and obtain several lower bounds for the spectral norm in terms of the ε -regularity theorem and the symplectic area of certain pseudoholomorphic curves. We then apply spectral invariants to the study of length minimizing properties of certain Hamiltonian paths among all paths. In addition to the construction of spectral invariants, these applications rely heavily on the chain-level Floer theory and on some existence theorems with energy bounds of pseudoholomorphic sections of certain Hamiltonian fibrations with prescribed monodromy. The existence scheme that we develop in this article in turn relies on some careful geometric analysis involving adiabatic degeneration and thick-thin decomposition of the Floer moduli spaces which has an independent interest of its own. We assume that ( M , ω ) is strongly semipositive throughout this article

Article information

Duke Math. J., Volume 130, Number 2 (2005), 199-295.

First available in Project Euclid: 15 November 2005

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53D35: Global theory of symplectic and contact manifolds [See also 57Rxx]
Secondary: 53D40: Floer homology and cohomology, symplectic aspects


Oh, Yong-Geun. Spectral invariants, analysis of the Floer moduli space, and geometry of the Hamiltonian diffeomorphism group. Duke Math. J. 130 (2005), no. 2, 199--295. doi:10.1215/00127094-8229689. https://projecteuclid.org/euclid.dmj/1132064627

Export citation


  • A. Banyaga, Sur la structure du groupe des difféomorphismes qui préservent une forme symplectique, Comment. Math. Helv. 53 (1978), 174,–,227.
  • M. Bialy and L. Polterovich, Geodesics of Hofer's metric on the group of Hamiltonian diffeomorphisms, Duke Math. J. 76 (1994), 273,–,292.
  • Yu. V. Chekanov, Lagrangian intersections, symplectic energy, and areas of holomorphic curves, Duke Math. J. 95 (1998), 213,–,226.
  • D. Dostoglou and D. A. Salamon, Self-dual instantons and holomorphic curves, Ann. of Math. (2) 139 (1994), 581,–,640.
  • M. Entov, K-area, Hofer metric and geometry of conjugacy classes in Lie groups, Invent. Math. 146 (2001), 93,–,141.
  • —, Commutator length of symplectomorphisms, Comment. Math. Helv. 79 (2004), 58,–,104.
  • M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 2003, no. 30, 1635,–,1676.
  • A. Floer, Morse theory for fixed points of symplectic diffeomorphisms, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 279,–,281.
  • —, The unregularized gradient flow of the symplectic action, Comm. Pure Appl. Math. 41 (1988), 775,–,813.
  • —, Symplectic fixed points and holomorphic spheres, Comm. Math. Phys. 120 (1989), 575,–,611.
  • K. Fukaya and Y.-G. Oh, Zero-loop open strings in the cotangent bundle and Morse homotopy, Asian J. Math. 1 (1997), 96,–,180.
  • K. Fukaya and K. Ono, Arnold conjecture and Gromov-Witten invariants, Topology 38 (1999), 933,–,1048.
  • M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307,–,347.
  • V. Guillemin, E. Lerman, and S. Sternberg, Symplectic Fibrations and Multiplicity Diagrams, Cambridge Univ. Press, Cambridge, 1996.
  • H. Hofer, On the topological properties of symplectic maps, Proc. Royal Soc. Edinburgh Sect. A 115 (1990), 25,–,38.
  • H. Hofer and D. A. Salamon, Floer homology and Novikov rings, Progr. Math. 133, Birkhaüser, Basel, 1995, 483,–,524.
  • H. Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977), 509,–,541.
  • F. Lalonde and D. Mcduff, The geometry of symplectic energy, Ann. of Math. (2) 141 (1995), 349,–,371.
  • F. Laudenbach, Engouffrement symplectique et intersections lagrangiennes, Comment. Math. Helv. 70 (1995), 558,–,614.
  • G. Liu and G. Tian, Floer homology and Arnold conjecture, J. Differential Geom. 49 (1998), 1,–,74.
  • D. Mcduff, Geometric variants of the Hofer norm, J. Symplectic Geom. 1 (2002), 197,–,252.
  • J. Milnor, Lectures on the h-Cobordism Theorem, Princeton Univ. Press, Princeton, 1965.
  • Y.-G. Oh, Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions, Comm. Pure Appl. Math. 45 (1992), 121,–,139.
  • —, Floer cohomology, spectral sequences, and the Maslov class of Lagrangian embeddings, Internat. Math. Res. Notices 1996, no. 7, 305,–,346.
  • —, Symplectic topology as the geometry of action functional, I: Relative Floer theory on the cotangent bundle, J. Differential Geom. 46 (1997), 499,–,577.
  • —, Symplectic topology as the geometry of action functional, II: Pants product and cohomological invariants, Comm. Anal. Geom. 7 (1999), 1,–,54.
  • —, Chain level Floer theory and Hofer's geometry of the Hamiltonian diffeomorphism group, Asian J. Math. 6 (2002), 579,–,624; Erratum, Asian J. Math. 7 (2003), 447,–,448. ${\!}$;
  • —, “Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds” in The Breadth of Symplectic and Poisson Geometry: Festschrift in Honor of Alan Weinstein, Progr. Math. 232, Birkhäuser, Boston, 2005, 525,–,570.
  • —, Normalization of the Hamiltonian and the action spectrum, J. Korean Math. Soc. 42 (2005), 65–83.
  • —, Spectral invariants and length minimizing property of Hamiltonian paths, Asian J. Math. 9 (2005), 1–18.
  • —, An existence theorem, with energy bounds, of Floer's perturbed Cauchy-Riemann equation with jumping discontinuity, preprint.
  • —, Floer mini-max theory, the Cerf diagram, and the spectral invariants, preprint.
  • —, Mini-max theory, spectral invariants and geometry of the Hamiltonian diffeomorphism group, preprint.
  • —, Thick and thin decompositions of the Floer moduli spaces and their applications, in preparation.
  • Y. Ostrover, A comparison of Hofer's metrics on Hamiltonian diffeomorphisms and Lagrangian submanifolds, Commun. Contemp. Math. 5 (2003), 803,–,811.
  • S. Piunikhin, D. Salamon, and M. Schwarz, “Symplectic Floer-Donaldson theory and quantum cohomology” in Contact and Symplectic Geometry (Cambridge, 1994), Publ. Newton. Inst. 8, Cambridge Univ. Press, Cambridge, 1996, 171,–,200.
  • L. Polterovich, The Geometry of the Group of Symplectic Diffeomorphisms, Lectures Math. ETH Zürich, Birkhäuser, Basel, 2001.
  • —, private communication, April 2003.
  • P. H. Rabinowitz, Periodic solutions of Hamiltonian systems, Comm. Pure Appl. Math. 31 (1978), 157,–,184.
  • Y. Ruan, “Virtual neighborhoods and pseudo-holomorphic curves” in Proceedings of the 6th Gökova Geometry-Topology Conference (Gökova, Turkey, 1998), Turkish J. Math. 23 (1999), 161,–,231.
  • J. Sacks and K. Uhlenbeck, The existence of minimal immersions of, $2$-spheres, Ann. of Math. (2) 113 (1981), 1,–,24.
  • D. Salamon and E. Zehnder, Morse theory for periodic solutions of Hamiltonian systems and the Maslov index, Comm. Pure Appl. Math. 45 (1992), 1303,–,1360.
  • M. Schwarz, On the action spectrum for closed symplectically aspherical manifolds, Pacific J. Math. 193 (2000), 419,–,461.
  • P. Seidel, $\pi_1$ of symplectic automorphism groups and invertibles in quantum homology rings, Geom. Funct. Anal. 7 (1997), 1046,–,1095.
  • K. Strebel, Quadratic Differentials, Ergeb. Math. Grenzgeb. (3) 5, Springer, Berlin, 1984.
  • C. Viterbo, Symplectic topology as the geometry of generating functions, Math. Ann. 292 (1992), 685,–,710.
  • B. Zwiebach, Closed string field theory: Quantum action and the Batalin-Vilkovisky master equation, Nuclear Phys. B 390 (1993), 33–152.