Communications in Mathematical Physics

, Volume 120, Issue 4, pp 575–611 | Cite as

Symplectic fixed points and holomorphic spheres

  • Andreas Floer
Article
Received:
Revised:

Abstract

LetP be a symplectic manifold whose symplectic form, integrated over the spheres inP, is proportional to its first Chern class. On the loop space ofP, we consider the variational theory of the symplectic action function perturbed by a Hamiltonian term. In particular, we associate to each isolated invariant set of its gradient flow an Abelian group with a cyclic grading. It is shown to have properties similar to the homology of the Conley index in locally compact spaces. As an application, we show that if the fixed point set of an exact diffeomorphism onP is nondegenerate, then it satisfies the Morse inequalities onP. We also discuss fixed point estimates for general exact diffeomorphisms.

Keywords

Manifold Abelian Group Action Function Compact Space Symplectic Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [A1]
    Arnold, V. I.: Sur une propriete topologique des applications globalement canoniques de la mecanique classique, C. R. Acad. Sci. Paris261, 3719–3722 (1965)Google Scholar
  2. [A2]
    —-: Mathematical methods of classical mechanics (Appendix 9), Nauka 1974; Engl. Transl. Berlin, Heidelberg, New York: Springer 1978Google Scholar
  3. [An]
    Aronszajn, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of the second order. J. Math. Pures Appl.36, 235–249 (1957)Google Scholar
  4. [AS1]
    Atiyah, M. F., Singer, I. M.: The index of elliptic operators I, III. Ann. Math.87, 484–530 (1968);87, 546–604 (1968)Google Scholar
  5. [AS2]
    ——, ——: Index theory of skew adjoint Fredholm operators. Publ. Math. IHES37, 305–325 (1969)Google Scholar
  6. [B]
    Banyaga, ——.: Sur la groupe des diffeomorphismes qui preservent une forme symplectique. Comment. Math. Helv.53, 174–227 (1978)Google Scholar
  7. [Bi]
    Birkhoff, G. D.: Proof of Poincaré's geometric theorem. Trans. AMS14, 14–22 (1912)Google Scholar
  8. [Ch]
    Chaperon, M.: Quelques questions de geometrie symplectique [d'apres, entre autres, Poincaré, Arnold, Conley et Zehnder], Seminaire Bourbaki 1982–83. Asterisque105–106, 231–249 (1983)Google Scholar
  9. [C]
    Conley, C. C.: Isolated invariant sets and the Morse index, CBMS Reg. Conf. Series in Math 38. Providence, RI: AMS 1978Google Scholar
  10. [CZ1]
    Conley, C. C., Zehnder, E.: The Birkhoff-Lewis fixed point theorem and a conjecture of V. I. Arnold. Invent. Math.73, 33–49 (1983)Google Scholar
  11. [CZ2]
    ——, ——: Morse type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure and Appl. Math.34, 207–253 (1984)Google Scholar
  12. [CZ3]
    ——, ——: Subharmonic solutions and Morse-theory. Physica124A, 649–658 (1984)Google Scholar
  13. [E]
    Eliashberg, Y. M.: Estimates on the number of fixed points of area preserving mappings. Preprint, : Syktyvkar University 1978Google Scholar
  14. [FHV]
    Floer, A., Hofer, H., Viterbo, C.: The Weinstein conjecture onP×ℂ, preprintGoogle Scholar
  15. [F1]
    Floer, A.: Proof of the Arnold conjecture for surfaces and generalizations to certain Kähler manifolds. Duke Math. J.53, 1–32 (1986)Google Scholar
  16. [F2]
    ——: A refinement of the Conley index and an application to the stability of hyperbolic invariant sets. Ergod. Theoret Dyn. Sys.7, 93–103 (1987)Google Scholar
  17. [F3]
    —-: Morse theory for Lagrangian intersections. J. Diff. Geom.28, (1988)Google Scholar
  18. [F4]
    —-: The unregularized gradient flow of the symplectic action. Commun. Pure Appl. Math. (to appear)Google Scholar
  19. [F5]
    —-: A relative Morse index for the symplectic action. Commun. Pure Appl. Math. (to appear)Google Scholar
  20. [F6]
    —-: Witten's complex and infinite dimensional Morse theory. J. Diff. Geom. (to appear)Google Scholar
  21. [F7]
    —-: Cuplength estimates for Lagrangian intersections (to appear)Google Scholar
  22. [F8]
    ——: An instanton-invariant for 3-manifolds. Commun. Math. Phys.118, 215–240 (1988)Google Scholar
  23. [Fo]
    Fortune, B.: A symplectic fixed point theorem for CPn. Invent. Math.81, 29–46 (1985)Google Scholar
  24. [Fr]
    Frnsoza, R.: Index filtrations and connection matrices for partially ordered Morse decompositions. PreprintGoogle Scholar
  25. [G]
    Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds. Invent. Math.82, 307–347 (1985)Google Scholar
  26. [Ho1]
    Hofer, H.: Lagrangian embeddings and critical point theory. Ann. Inst. H. Poincaré. Analyse non lineaire2, 407–462 (1985)Google Scholar
  27. [HoZ]
    Hofer, H.: Ljusternik-Snirelman theory for Lagrangian intersections, preprintGoogle Scholar
  28. [H]
    Hörmander, L.: The analysis of linear differential operators III. Berlin, Heidelberg, New York: Springer 1985Google Scholar
  29. [Hu]
    Husemoller, D.: Fibre bundles, Springer Grad. Texts in Math. vol.20. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  30. [Ka]
    Kawabe, H.: A symplectic fixed point theorem in case that a symplectic form is exact in the universal covering space and a manifold has no conjugate points, Preprint, Tokyo Inst. of Tech. 1987Google Scholar
  31. [Kl]
    Klingenberg, W.: Lectures on closed geodesics. Grundl. der math. Wiss vol.230. Berlin, Heidelberg, New York: Springer 1978Google Scholar
  32. [Ko]
    Kondrat'ev, V. A.: Boundary value problems for elliptic equations in domains with conical or angular point. Trans. Mosc. Math. Soc.16, (1967)Google Scholar
  33. [Ku]
    Kuiper, N. H.: The homotopy type of the unitary group of Hilbert space. Topology3, 19–30 (1965)Google Scholar
  34. [LS]
    Laudenbach, F., Sikorav, J. C.: Persistance d'intersection avec la section nulle au cours d'une isotopie hamiltonienne dans un fibre cotangent. Invent. Math.82, 349–357 (1985)Google Scholar
  35. [LM]
    Lockhard, R. B., McOwen, R. C.: Elliptic operators on noncompact manifolds. Ann. Sci. Norm. Sup. PisaIV-12, 409–446 (1985)Google Scholar
  36. [MW]
    Marsden, J., Weinstein, A.: Reduction of symplectic manifolds with symmetry. Rep. Math. Phys.5, 121–130 (1974)Google Scholar
  37. [MP]
    Maz'ja, V. G., Plamenevski, B. A.: Estimates onL pand Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary problems in domans with singular points on the boundary. Math. Nachr.81, 25–82 (1978) (in Russian), English Transl. In: AMS Trans., Ser. 2,123, 1–56 (1984)Google Scholar
  38. [McD]
    McDuff, D.: Examples of symplectic structures. Invent. Math.89, 13–36 (1987)Google Scholar
  39. [M]
    Milnor, J.: Lectures on the h-cobordism theorem. Math. Notes, Princeton, NJ: Princeton Univ. Press 1965Google Scholar
  40. [N]
    Nikishin, N.: Fixed points of diffeomorphisms on the two sphere that preserve area. Funk. Anal. i Prel.8, 84–85 (1984)Google Scholar
  41. [Ps]
    Palais, R. S.: Morse theory on Hilbert manifolds. Topology2, 299–340 (1963)Google Scholar
  42. [Pu]
    Pansu, P.: Sur l'article de M. Gromov, Preprint, Ecole Polytechnique, Palaiseau 1986Google Scholar
  43. [P]
    Poincaré, H.: Sur une theoreme de geometrie, Rend. Circolo Mat. Palermo33, 375–407 (1912)Google Scholar
  44. [Q]
    Qinn, F.: Transversal approximation on Banach manifolds. In: Proc. Symp. Pure Math. vol.15. Providence, RI: AMS 1970Google Scholar
  45. [Ra]
    Rabinowitz, P. H.: Periodic solutions of Hamiltonian systems. Commun. Pure Appl. Math.31, 336–352 (1979)Google Scholar
  46. [Ry]
    Rybakowski, K. P.: The homotopy index and partial differential equations, Universitext. Berlin, Heidelberg, New York: Springer 1987Google Scholar
  47. [RZ]
    Rybakowski, K. P., Zehnder, E.: A Morse equation in Conley's index theory for semiflows on metric spaces. Ergod. Theoret Dyn. Sys.5, 123–143 (1985)Google Scholar
  48. [SU]
    Sachs, J., Uhlenbeck, K. K.: The existence of minimal 2-spheres. Ann. Math.113, 1–24 (1981)Google Scholar
  49. [S1]
    Sikorav, J. C.: Points fixes d'un symplectomorphisme homologue a l'identite. J. Diff. Geom.22, 49–79 (1982)Google Scholar
  50. [S2]
    —-: Homologie de Novikov associee a une classe de cohomologie reelle de degree un these, Orsay, 1987Google Scholar
  51. [Si]
    Simon, C. P.: A bound for the fixed point index of an area preserving map with applications to mechanics. Invent. Math.26, 187–200 (1974)Google Scholar
  52. [Sm]
    Smale, S.: An infinite dimensional version of Sard's theorem. Am. J. Math.87, 213–221 (1973)Google Scholar
  53. [Sp]
    Spanier, E.: Algebraik topology. New York: McGraw-Hill 1966Google Scholar
  54. [T1]
    Taubes, C. H.: Self-dual connections on manifolds with indefinite intersection matrix. J. Diff. Geom.19, 517–5670 (1984)Google Scholar
  55. [T2]
    ——: Gauge theory on asymptotically periodic 4-manifolds. J. Diff. Geom.25, 363–430 (1987)Google Scholar
  56. [V]
    Viterbo, C.: Intersections de sous-varietes lagrangiennes, fonctionelles d'action et indice des systemes hamiltoniens. Preprint 1986Google Scholar
  57. [W1]
    Weinstein, A.: Periodic orbits for convex Hamiltonian systems. Ann. Math.108, 507–518 (1978)Google Scholar
  58. [W2]
    ——: On extending the Conley Zehnder fixed point theorem to other manifolds. Proc. Symp. Pure Math. vol.45. Providence, R. J.: AMS 1986Google Scholar
  59. [Wi]
    Witten, E.: Supersymmetry and Morse theory. J. Diff. Geom.17, 661–692 (1982)Google Scholar
  60. [Wo]
    Wolfson, J. G.: A P.D.E. proof of Gromov's compactness of pseudoholomorphic curves, preprint, Tulane University 1986Google Scholar
  61. [Z]
    Zehnder, E.: Periodic solutions of Hamiltonian equations. In: Lecture Notes in Mathematics, vol.1031. pp, 172–1213 Berlin, Heidelberg, New York: Springer 1983Google Scholar

Copyright information

© Springer-Verlag 1989

Authors and Affiliations

  • Andreas Floer
    • 1
  1. 1.Courant InstituteNew YorkUSA

Personalised recommendations