Bounds on spectral norms and barcodes

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.

71 pages, 6 figures

References in corpus (7)

• Quantum Structures for Lagrangian Submanifolds
• Moduli of J-Holomorphic Curves with Lagrangian Boundary Conditions and Open Gromov-Witten Invariants for an $S^1$-Equivariant Pair
• The action spectrum and C^0 symplectic topology
• The Lagrangian Floer-quantum-PSS package and canonical orientations in Floer theory
• Quasi-morphisms and quasi-states in symplectic topology
• Barcodes and area-preserving homeomorphisms
• Contact spectral invariants and persistence

Cited by in corpus (11)

• The action spectrum and C^0 symplectic topology
• On $C^0$-continuity of the spectral norm for symplectically non-aspherical manifolds
• Hamiltonian no-torsion
• Topological Persistence in Geometry and Analysis
• Completeness of derived interleaving distances and sheaf quantization of non-smooth objects
• A comparison of categorical and topological entropies on Weinstein manifolds
• Approximate Identities and Lagrangian Poincaré Recurrence
• The spectral diameter of a Liouville domain
• Dehn-Seidel twist, $C^0$ symplectic topology and barcodes
• Lagrangian intersections and a conjecture of Arnol'd
• Approximation of Generating Function Barcode for Hamiltonian Diffeomorphisms