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Spectrally-large scale geometry in cotangent bundles

Qi Feng, Jun Zhang

Abstract

In this paper, we prove that the ${\rm Ham}$-orbit space from a fiber of a large family of cotangent bundles, as a metric space with respect to the Floer-theoretic spectral metric, contains a quasi-isometric embedding of an infinite-dimensional normed vector space. The same conclusion holds for the group of compactly supported Hamiltonian diffeomorphisms of some cotangent bundles. To prove this, we generalize a result, relating boundary depth and spectral norm for closed symplectic manifolds in Kislev-Shelukhin's recent work, to Liouville domains. Then we modify Usher's constructions (which were used to obtain Hofer-large scale geometric properties) to achieve our desired conclusions.

Spectrally-large scale geometry in cotangent bundles

Abstract

In this paper, we prove that the -orbit space from a fiber of a large family of cotangent bundles, as a metric space with respect to the Floer-theoretic spectral metric, contains a quasi-isometric embedding of an infinite-dimensional normed vector space. The same conclusion holds for the group of compactly supported Hamiltonian diffeomorphisms of some cotangent bundles. To prove this, we generalize a result, relating boundary depth and spectral norm for closed symplectic manifolds in Kislev-Shelukhin's recent work, to Liouville domains. Then we modify Usher's constructions (which were used to obtain Hofer-large scale geometric properties) to achieve our desired conclusions.
Paper Structure (15 sections, 14 theorems, 132 equations, 5 figures, 1 table)

This paper contains 15 sections, 14 theorems, 132 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Method of proofs
  4. Summary of current art
  5. Various ingredients in Floer theories
  6. Hamiltonian Floer theory
  7. Wrapped Floer cohomology
  8. Product structure of wrapped Floer cohomology
  9. PSS morphism
  10. Proofs
  11. Proof of Theorem \ref{['betaleqgamma']}
  12. Proof of Theorem \ref{['Ham-infinite-flat']}
  13. Proof of Theorem \ref{['Lag-infinite-flat']}
  14. Proof of Proposition \ref{['sphere-flat']}
  15. Proof of Theorem \ref{['module-structure']}

Key Result

Theorem A

Let $(N, g)$ be a closed Riemannian manifold. Suppose there is no non-constant contractible closed geodesic in $(N,g)$, then metric space $\left(\mathop{\mathrm{Ham}}\nolimits(D_g^*N,\omega_{\rm can}),d_{\gamma}\right)$ contains a rank-$\infty$ quasi-flat.

Figures (5)

  • Figure 1: A strip with a slit.
  • Figure 2: The desired Hamiltonian $1$-form $K$.
  • Figure 3: Compactification of $\mathcal{M}_4$.
  • Figure 4: The image of $\phi_H^1(F_x)$, and bi-gons from $x_2, x_3$ to $x_1$.
  • Figure 5: The module structure.

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem A
  • Conjecture 1.2
  • Theorem B
  • Example 1.3
  • Theorem C
  • Definition 1.4
  • Example 1.5
  • Theorem D
  • Theorem 1.6
  • ...and 22 more