Mathematische Annalen

, Volume 341, Issue 1, pp 1–13 | Cite as

Spectral flexibility of symplectic manifolds T 2 × M

Article

Abstract

We consider Riemannian metrics compatible with the natural symplectic structure on T 2 × M, where T 2 is a symplectic 2-torus and M is a closed symplectic manifold. To each such metric we attach the corresponding Laplacian and consider its first positive eigenvalue λ1. We show that λ1 can be made arbitrarily large by deforming the metric structure, keeping the symplectic structure fixed. The conjecture is that the same is true for any symplectic manifold of dimension  ≥ 4. We reduce the general conjecture to a purely symplectic question.

Mathematics Subject Classification (2000)

35P15 53D05 53C17 

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References

  1. 1.
    Adams, R.: Sobolev spaces. In: Pure and Applied Mathematics, vol. 65. Academic [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1975)Google Scholar
  2. 2.
    Adams R., Aronszajn N. and Smith K.T. (1967). Theory of Bessel potentials. II. Ann. Inst. Fourier (Grenoble) 17: 1–135 MATHMathSciNetGoogle Scholar
  3. 3.
    Adams R., Aronszajn N. and Hanna M.S. (1969). Theory of Bessel potentials. Part III. Potentials on regular manifolds. Ann. Inst. Fourier (Grenoble) 19: 279–338 MATHMathSciNetGoogle Scholar
  4. 4.
    Aronszajn N., Mulla F. and Szeptycki P. (1963). On spaces of potentials connected with L p classes. Ann. Inst. Fourier (Grenoble) 13: 211–306 MATHMathSciNetGoogle Scholar
  5. 5.
    Aronszajn N. and Smith K.T. (1961). Theory of Bessel potentials. I. Ann. Inst. Fourier (Grenoble) 11: 385–475 MATHMathSciNetGoogle Scholar
  6. 6.
    Bérard-Bergery L. and Bourguignon J.-P. (1982). Laplacians and Riemannian submersions with totally geodesic fibres. Illinois J. Math. 26: 181–200 MATHMathSciNetGoogle Scholar
  7. 7.
    Bourguignon J.-P., Li P. and Yau S.T. (1994). Upper bound for the first eigenvalue of algebraic submanifolds. Comment. Math. Helv. 69: 199–207 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Calderón, A.P.: Lebesgue spaces of differentiable functions and distributions. In: Proceedings of the Symposium on Pure Mathematics, vol. IV, pp. 33–49. American Mathematical Society, Providence (1961)Google Scholar
  9. 9.
    Colbois B. and Dodziuk J. (1994). Riemannian metrics with large λ1. Proc. Am. Math. Soc. 122: 905–906 MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Demailly, J.-P.: L 2 vanishing theorems for positive line bundles and adjunction theory. In: Transcendental methods in algebraic geometry (Cetraro, 1994). Lecture Notes in Mathematics, vol. 1646, pp. 1–97. Springer, Berlin (1996)Google Scholar
  11. 11.
    Edmunds D.E. and Triebel H. (1996). Function Spaces, Entropy Numbers, Differential Operators, Cambridge Tracts in Mathematics, vol. 120. Cambridge University Press, Cambridge Google Scholar
  12. 12.
    El Soufi, A., Ilias, S.: Le volume conforme et ses applications d’après Li et Yau. Séminaire de Théorie Spectrale et Géométrie, Année 1983–1984, pp. VII.1–VII.15. Univ. Grenoble I, Saint (1984)Google Scholar
  13. 13.
    Friedlander L. and Nadirashvili N. (1999). A differential invariant related to the first eigenvalue of the Laplacian. Int. Math. Res. Notices 17: 939–952 CrossRefMathSciNetGoogle Scholar
  14. 14.
    Hersch J. (1970). Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér. A-B 270: A1645–A1648 MathSciNetGoogle Scholar
  15. 15.
    McDuff D. and Salamon D. (1995). Introduction to Symplectic Topology, Oxford Mathematical Monographs. Clarendon/Oxford University Press, New York Google Scholar
  16. 16.
    Polterovich L. (1998). Symplectic aspects of the first eigenvalue. J. Reine Angew. Math. 502: 1–17 MATHMathSciNetGoogle Scholar
  17. 17.
    Rothschild L.P. and Stein E.M. (1976). Hypoelliptic differential operators and nilpotent groups. Acta Math. 137: 247–320 CrossRefMathSciNetGoogle Scholar
  18. 18.
    Stein E.M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series. Princeton University Press, Princeton Google Scholar
  19. 19.
    Triebel H. (1983). Theory of Function Spaces, Monographs in Mathematics, vol. 78. Birkhäuser, Basel Google Scholar
  20. 20.
    Triebel H. (1992). Theory of Function Spaces. II, Monographs in Mathematics, vol. 84. Birkhäuser, Basel Google Scholar
  21. 21.
    Yang P.C. and Yau S.T. (1980). Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7(4): 55–63 MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.IHESBures-sur-YvetteFrance

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