Hofer’s metrics and boundary depth

Abstract

We show that if (M , ω) 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 , ω) 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 × 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.1991 Mathematics Subject Classification. 53D22, 53D40.

$p$-cyclic persistent homology and Hofer distance

“…We simply remark here that almost all the ingredients above can be defined and constructed in a parallel way by starting from a covering space of α (X ) once we fixed a reference non-contractible loop in the homotopy class α. The general construction has been carried out explicitly in Section 5 in [Ush13]. Proposition 5.1 in [Ush13] implies that Theorem 2.4 still holds for C F * (H, J) α .…”

A Quasi-isometric embedding into the group of Hamiltonian diffeomorphisms with Hofer’s metric

“…The boundary depth of a Hamiltonian diffeomorphism is our main motivation for studying barcodes, so we pause very briefly to remind the reader of its definition, its relation to barcodes, and a few of its key properties. See [25], [27], and [28] for more details.…”

“…Provided the existence of a closed Lagrangian L ⊂ M which admits a Riemannian metric of non-positive curvature and has the inclusion-induced map i * : π 1 (L) → π 1 (M ) injective, Py shows that for any m ∈ there exists an embedding φ : m → H am(M , ω) and a constant C m > 0 satisfyingThis result was generalized in [27], in which Usher proves that if M admits an autonomous Hamiltonian H : M → whose flow has all of its contractible periodic orbits constant, then there exists an embedding of ∞ into H am(M , ω) similar to the one presented in [20]. It should be noted that Py's assumptions imply the existence of such an H, as explained in [27].While the conclusion of Theorem 1.1 is much weaker than Usher's result and somewhat weaker than that of Py, its assumptions are quite mild and indeed do not lie entirely within the scope of these previous results. For instance, Usher points out in [27] that any closed toric manifold M will not admit an autonomous Hamiltonian H as described in the previous paragraph, and so any such manifold which is also (negative) monotone (for example, (S 2 , ω)) is one for which Theorem 1.1 asserts something new about the geometry of (H am(M , ω), d H ).The Hamiltonian diffeomorphisms which define our embedding are generated by radially symmetric functionsF i which are zero outside of B(2πR) and of the formf i |z| 2 2 , withf i : [0, R] → , for z ∈ B(2πR).…”

“…Similarly, one may instead ask the broader question of which H am(M , ω) admit quasi-isometric embeddings of multi-dimensional normed vector spaces. This question already has partial answers, among which are results appearing in [20] and [27]. Provided the existence of a closed Lagrangian L ⊂ M which admits a Riemannian metric of non-positive curvature and has the inclusion-induced map i * : π 1 (L) → π 1 (M ) injective, Py shows that for any m ∈ there exists an embedding φ : m → H am(M , ω) and a constant C m > 0 satisfyingThis result was generalized in [27], in which Usher proves that if M admits an autonomous Hamiltonian H : M → whose flow has all of its contractible periodic orbits constant, then there exists an embedding of ∞ into H am(M , ω) similar to the one presented in [20].…”

“…This question already has partial answers, among which are results appearing in [20] and [27]. Provided the existence of a closed Lagrangian L ⊂ M which admits a Riemannian metric of non-positive curvature and has the inclusion-induced map i * : π 1 (L) → π 1 (M ) injective, Py shows that for any m ∈ there exists an embedding φ : m → H am(M , ω) and a constant C m > 0 satisfyingThis result was generalized in [27], in which Usher proves that if M admits an autonomous Hamiltonian H : M → whose flow has all of its contractible periodic orbits constant, then there exists an embedding of ∞ into H am(M , ω) similar to the one presented in [20]. It should be noted that Py's assumptions imply the existence of such an H, as explained in [27].While the conclusion of Theorem 1.1 is much weaker than Usher's result and somewhat weaker than that of Py, its assumptions are quite mild and indeed do not lie entirely within the scope of these previous results.…”