Lagrangian metric geometry with Riemannian bounds

Chassé, Jean-Philippe

doi:10.2140/agt.2026.26.1923

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

Submission form

Editorial board

Subscriptions

ISSN (electronic): 1472-2739

ISSN (print): 1472-2747

Author index

To appear

Other MSP journals

Lagrangian metric geometry with Riemannian bounds

Jean-Philippe Chassé

Algebraic & Geometric Topology 26 (2026) 1923–1963

DOI: 10.2140/agt.2026.26.1923

Abstract

We study collections of exact Lagrangian submanifolds respecting some uniform Riemannian bounds, which we equip with a metric naturally arising in symplectic topology (e.g., the Lagrangian Hofer metric or the spectral metric). We exhibit many metric and symplectic properties of these spaces, such that they have compact completions and that they contain only finitely many Hamiltonian isotopy classes. We then use this to exclude many unusual phenomena from happening in these bounded spaces. Taking limits in the bounds, we also conclude that there are at most countably many Hamiltonian isotopy classes of exact Lagrangian submanifolds in a Liouville manifold. Under some mild topological assumptions, we get analogous results for monotone Lagrangian submanifolds with a fixed monotonicity constant. Finally, in the process of showing these results, we get new results on the Riemannian geometry of cotangent bundles and surfaces which might be of independent interest.

Keywords

Lagrangian submanifolds, Hausdorff metric, Hofer metric, spectral metric, shadow metrics, Riemannian geometry, Sasaki metric

Mathematical Subject Classification

Primary: 53D12

Secondary: 51F99, 53C17, 53C22

References

Publication

Received: 1 August 2024

Revised: 16 May 2025

Accepted: 14 June 2025

Published: 23 May 2026

Authors

Jean-Philippe Chassé
CRM Université de Montréal C.P. 6128 Succ. Centre-Ville, Montréal, QC H3C 3J7 Canada

Open Access made possible by participating institutions via Subscribe to Open.

Volume 26, issue 5 (2026)