Volume 25, issue 7 (2021)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 26
Issue 3, 937–1434
Issue 2, 477–936
Issue 1, 1–476

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Interests
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
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