Spectral Flexibility of Symplectic Manifolds T 2 ×

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. MSC: 35P15; 53D05; 53C17