UNBOUNDEDNESS OF THE FIRST EIGENVALUE OF THE LAPLACIAN IN THE SYMPLECTIC CATEGORY

Abstract

Given a closed symplectic manifold (M2n, ω) of dimension 2n ≥ 4, we consider all Riemannian metrics on M, which are compatible with the symplectic structure ω. For each such metric g, we look at the first eigenvalue λ1 of the Laplacian associated with it. We show that λ1 can be made arbitrarily large, when we vary g. This generalizes previous results of Polterovich, and of Mangoubi.