Lagrangian metric geometry with Riemannian bounds

TL;DR

This paper investigates the metric geometry of collections of exact Lagrangian submanifolds with uniform Riemannian bounds, revealing properties like compactness, finiteness of isotopy classes, and countability results in symplectic topology.

Contribution

It introduces a framework for analyzing Lagrangian submanifolds with Riemannian bounds using symplectic metrics, establishing new compactness and classification results.

Findings

Spaces have compact completions

Finitely many Hamiltonian isotopy classes in bounded spaces

Countably many isotopy classes in Liouville manifolds

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.