Unboundedness of the first eigenvalue of the Laplacian in symplectic category

TL;DR

This paper proves that on any closed symplectic manifold of dimension greater than 2, the first eigenvalue of the Laplacian can be made arbitrarily large by varying compatible Riemannian metrics, extending previous results.

Contribution

It establishes the unboundedness of the first Laplacian eigenvalue in the symplectic category, generalizing earlier work by Polterovich and Mangoubi.

Findings

First eigenvalue  can be arbitrarily large for compatible metrics.

Results apply to all closed symplectic manifolds of dimension > 2.

Extends previous unboundedness results to a broader class of manifolds.

Abstract

Given a closed symplectic manifold (M,\omega) of dimension greater than 2, we consider all Riemannian metrics on M, which are compatible with the symplectic structure \omega. For each such metric, we look at the first eigenvalue \lambda_1 of the Laplacian associated with it. We show that \lambda_1 can be made arbitrarily large, when we vary the metric. This generalizes previous results of Polterovich, and of Mangoubi.