Volume 25, issue 7 (2021)

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Bounds on spectral norms and barcodes

Asaf Kislev and Egor Shelukhin

Geometry & Topology 25 (2021) 3257–3350

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 nondegeneracy 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.

Hamiltonian diffeomorphisms, Lagrangian submanifolds, Floer homology, persistence modules, barcodes, spectral norm, spectral invariants
Mathematical Subject Classification 2010
Primary: 57R17
Secondary: 53D12, 53D40
Received: 2 November 2018
Revised: 3 July 2020
Accepted: 23 August 2020
Published: 25 January 2022
Proposed: Yasha Eliashberg
Seconded: Paul Seidel, András I Stipsicz
Asaf Kislev
School of Mathematical Sciences
Tel Aviv University
Tel Aviv
Egor Shelukhin
Department of Mathematics and Statistics
University of Montreal
Montreal, QC