Norms of Modes and Quasi-modes Revisited

This article is devoted to the analysis of eigenfunctions (modes) and approximate eigenfunctions (quasi-modes) of the Laplacian on a compact manifold (M, g) with completely integrable geodesic flow. We give a new proof of the main result of [TZ] that (M, g) with integrable Laplacians and with uniformly bounded eigenfunctions must be flat. The proof is based on the use of Birkhoff normal forms and on a comparison of modes and quasimodes. In the process, we discuss tunnelling between resonant tori and give a proof that eigenfunctions concentrate on individual tori in the non-resonant case. We also give brief expositions of results in [TZ2, SZ].