Bounds on spectral norms and barcodes

TL;DR

This paper establishes bounds on spectral norms related to Lagrangian Floer homology, linking algebraic invariants to persistence modules, and introduces new methods for bounding and calculating these norms in symplectic geometry.

Contribution

It introduces a novel averaging method for spectral norm bounds and applies persistence module theory to prove sharp bounds and compute Lagrangian quantum homology.

Findings

Spectral norm controls the barcode of Hamiltonian perturbations.

New averaging method provides uniform bounds in specific symplectic manifolds.

Bounds are proven to be sharp in certain cases.

Abstract

We investigate the relations between algebraic structures, spectral invariants, and persistence modules, in the context of monotone Lagrangian Floer homology with Hamiltonian term. Firstly, we use the newly introduced method of filtered continuation elements to prove that the Lagrangian spectral norm controls the barcode of the Hamiltonian perturbation of the Lagrangian submanifold, up to shift, in the bottleneck distance. Moreover, we show that it satisfies Chekanov type low-energy intersection phenomena, and non-degeneracy theorems. Secondly, we introduce a new averaging method for bounding the spectral norm from above, and apply it to produce precise uniform bounds on the Lagrangian spectral norm in certain closed symplectic manifolds. Finally, by using the theory of persistence modules, we prove that our bounds are in fact sharp in some cases. Along the way we produce a new calculation of the Lagrangian quantum homology of certain Lagrangian submanifolds, and answer a question of Usher.