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

Abstract

We construct an embedding Φ of [0, 1] ∞ into H am (M , ω), the group of Hamiltonian diffeomorphisms of a suitable closed symplectic manifold (M , ω). We then prove that Φ is in fact a quasi-isometry. After imposing further assumptions on (M , ω), we adapt our methods to construct a similar embedding of ⊕ [0, 1] ∞ into either H am (M , ω) or H am (M , ω), the universal cover of H am (M , ω). Along the way, we prove results related to the filtered Floer chain complexes of radially symmetric Hamiltonians. Our proofs rely heavily on a continuity result for barcodes (as presented in [28]) associated to filtered Floer homology viewed as a persistence module. CONTENTS1. Introduction 1 2. Hamiltonian Floer homology 4 3. Barcodes 7 3.1. Persistence modules and barcodes 7 3.2. Boundary depth 8 4. Radially symmetric Hamiltonians 8 5. Proof of the main theorem 20 6. Adaptation: Embeddings of ⊕ [0, 1] ∞ 32 References 35Hofer's metric d H on H am(M , ω) is then defined as d H (φ, ψ) = ||φ −1 • ψ|| H for any φ, ψ ∈ H am(M , ω). Now let [0, 1] ∞ denote the set of all [0, 1]-valued sequences with only finitely many non-zero entries, and for a = {a iWith these notations established, we may state our main theorem as follows.Theorem 1.1. Let M be a closed symplectic manifold which is either monotone or negative monotone. Suppose we may symplectically embed a ball B(2πR) of radius 2R into M , where if M 's rationality constant γ is non-zero, we require 4πR ≤ γ. Then for any ǫ > 0, there exists an embedding Φ :, and the rationality constant γ of M is the non-negative generator of the image of [ω]| π 2 (M ) . In particular, this paper has N = 0, γ = 0 when c 1 (T M )| π 2 (M ) = 0, [ω]| π 2 (M ) = 0, respectively; while this breaks from the convention of setting N , γ equal to ∞ in such cases, we find it to be a worthwhile one for this work as it simplifies the discussion of the various cases considered in the proof of Theorem 1.1 (particularly Lemma 5.2). Upon the introduction of Hofer's metric, it was natural to ask which symplectic manifolds (M , ω) yield (H am(M , ω), d H ) with infinite diameter, and the appearances of results in this direction form a rich history; see [11], [17], [23], [15], [4], and [12], for example. 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]. It should be noted that Py's...

Bounds on spectral norms and barcodes

“…As a corollary of Proposition 15, using variations on the computations of the Lagrangian Seidel element in [61], we obtain uniform bounds on γ(L, φ 1 H ) and β(φ 1 H ) in the following four cases. Moreover, adapting arguments of Stevenson [108], we show that in some of these cases, our bounds are the best ones possible, uniform in H. This provides a class of examples with finite β(M, L) and provides a more precise answer to Usher's question.…”

“…[12,22,23,28,31,34,36,49,121]). Recently, persistence modules found applications in symplectic topology, see for example [6,42,93,94,108,116,120], with precursors in [10,32,46,113,114].…”

“…The goal of this section is to give examples of Lagrangian submanifolds L ⊂ M with large boundary depths β(M, L). The main reference for this section is [108], where examples for radially symmetric Hamiltonian diffeomorphisms with large boundary depths are constructed. Here we work out variants of these examples in the relative case.…”

Viterbo conjecture for Zoll symmetric spaces

“…This map is Lipschitz with respect to the L 1,∞ -distance on H = H M , and the bottleneck distance on the space barcodes of barcodes. This observation was used in [76], in [3,40,78,92,99,105] and more recently in [18,31,60,66,93,95] to produce various quantitative results in symplectic topology. Set barcodes ′ for the quotient space of barcodes with respect to the isometric R-action by shifts.…”

Persistence Modules with Operators in Morse and Floer Theory

“…[8,9,12,14,15,17,20,34,62]). Quite recently, persistence modules found applications in symplectic topology, see [1,29,47,52,56,61], with preludes in [6,31,54,55].…”