Hofer's metrics and boundary depth

TL;DR

This paper demonstrates that under certain dynamical conditions on a closed symplectic manifold, Hofer's metric exhibits infinite diameter and contains infinite-dimensional quasi-isometric embeddings, using Floer theory's boundary depth.

Contribution

It establishes a link between the dynamical properties of Hamiltonian vector fields and the large-scale geometry of Hofer's metric, introducing boundary depth as a key tool.

Findings

Hofer's metric has infinite diameter under specified conditions.

The metric admits infinite-dimensional quasi-isometric embeddings.

Boundary depth encodes robust symplectic-topological information.

Abstract

We show that if (M,\omega) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer's metric on the group of Hamiltonian diffeomorphisms of (M,\omega) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer's metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in M x M when M satisfies the above dynamical condition. To prove this, we use the properties of a Floer-theoretic quantity called the boundary depth, which measures the nontriviality of the boundary operator on the Floer complex in a way that encodes robust symplectic-topological information.