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.
Similar content being viewed by others
References
Adams, R.: Sobolev spaces. In: Pure and Applied Mathematics, vol. 65. Academic [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York (1975)
Adams R., Aronszajn N. and Smith K.T. (1967). Theory of Bessel potentials. II. Ann. Inst. Fourier (Grenoble) 17: 1–135
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
Aronszajn N., Mulla F. and Szeptycki P. (1963). On spaces of potentials connected with L p classes. Ann. Inst. Fourier (Grenoble) 13: 211–306
Aronszajn N. and Smith K.T. (1961). Theory of Bessel potentials. I. Ann. Inst. Fourier (Grenoble) 11: 385–475
Bérard-Bergery L. and Bourguignon J.-P. (1982). Laplacians and Riemannian submersions with totally geodesic fibres. Illinois J. Math. 26: 181–200
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
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)
Colbois B. and Dodziuk J. (1994). Riemannian metrics with large λ1. Proc. Am. Math. Soc. 122: 905–906
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)
Edmunds D.E. and Triebel H. (1996). Function Spaces, Entropy Numbers, Differential Operators, Cambridge Tracts in Mathematics, vol. 120. Cambridge University Press, Cambridge
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)
Friedlander L. and Nadirashvili N. (1999). A differential invariant related to the first eigenvalue of the Laplacian. Int. Math. Res. Notices 17: 939–952
Hersch J. (1970). Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér. A-B 270: A1645–A1648
McDuff D. and Salamon D. (1995). Introduction to Symplectic Topology, Oxford Mathematical Monographs. Clarendon/Oxford University Press, New York
Polterovich L. (1998). Symplectic aspects of the first eigenvalue. J. Reine Angew. Math. 502: 1–17
Rothschild L.P. and Stein E.M. (1976). Hypoelliptic differential operators and nilpotent groups. Acta Math. 137: 247–320
Stein E.M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series. Princeton University Press, Princeton
Triebel H. (1983). Theory of Function Spaces, Monographs in Mathematics, vol. 78. Birkhäuser, Basel
Triebel H. (1992). Theory of Function Spaces. II, Monographs in Mathematics, vol. 84. Birkhäuser, Basel
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