Inverse reduction inequalities for spectral numbers and applications

TL;DR

This paper establishes a new inequality relating spectral numbers of Lagrangians and their reductions, with implications for spectral boundedness conjectures and the structure of Hamiltonian diffeomorphisms.

Contribution

It proves an inverse inequality for spectral numbers of Lagrangians and their reductions, advancing understanding in symplectic topology.

Findings

Proved an inequality between spectral numbers of Lagrangians and their reductions.

Applied results to the spectral boundedness conjecture for Lagrangians.

Analyzed the local path-connectedness of Hamiltonian diffeomorphisms with the spectral metric.

Abstract

Our main result is the proof of an inequality between the spectral numbers of a Lagrangian and the spectral numbers of its reductions, in the opposite direction to the classical inequality (see e.g [Vit92]). This has applications to the "Geometrically bounded Lagrangians are spectrally bounded" conjecture from [Vit08], to the structure of elements in the $\gamma$-completion of the set of exact Lagrangians. We also investigate the local path-connectedness of the set of Hamiltonian diffeomorphisms with the spectral metric.