Spectral Flexibility of Symplectic Manifolds T^2 x M

Abstract

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.