A ug 2 00 5 Spectral Flexibility of the Symplectic Manifold T 2 ×

We consider Riemannian metrics compatible with the 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. This extends a theorem of L. Polterovich on T 4 ×M . The conjecture is that the same is true for any symplectic manifold of dimension ≥ 4.