Spectral Flexibility of Symplectic Manifolds T^2 x M

Details Spectral Flexibility of Symplectic Manifolds T^2 x M Spectral Flexibility of Symplectic Manifolds T^2 x M

2005-08-07

arxiv.org/abs/math/0508128v3

Math. Ann. 341 (2008), no. 1, 1--13

Mathematics

Spectral Theory

15 Pages; introduction revised; to appear in Math. Ann

Scientific paper

10.1007/s00208-007-0178-z

We consider Riemannian metrics compatible with the natural symplectic structure on T^2 x 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 \lambda_1. We show that \lambda_1 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.

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