Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectrally-large scale geometry via set-heaviness

Qi Feng, Jun Zhang

TL;DR

The paper investigates when the spectral norm on the Hamiltonian diffeomorphism group of a Liouville domain yields unbounded large-scale geometry, linking this to the non-vanishing of symplectic cohomology via heaviness and heavy hypersurfaces. It develops a framework unifying Liouville-domain and closed-manifold settings, showing that $\mathrm{SH}^*(W,\omega)\neq 0$ implies the existence of rank-$\infty$ quasi-flats in $(\mathrm{Ham}(W,\omega), d_{\gamma})$, and extends the theory to Lagrangian orbit spaces using wrapped Floer cohomology with $L$-heavy notions. The work outlines two main strategies—boundary-depth-based embeddings and constructions from heavy subsets—along with alternative approaches using egg-beater models and homogenized quasi-morphisms, yielding a robust dichotomy: either the spectral-norm geometry is bounded or it contains rank-$\infty$ flats. These results apply to unit co-disk bundles and other Liouville domains, revealing deep connections between symplectic cohomology, heaviness, and large-scale geometry in both absolute and relative (Lagrangian) settings.

Abstract

We show that there exist infinite-dimensional quasi-flats in the compactly supported Hamiltonian diffeomorphism group of the Liouville domain, with respect to the spectral norm, if and only if the symplectic cohomology of this Liouville domain does not vanish. In particular, there exist infinite-dimensional quasi-flats in the compactly supported Hamiltonian diffeomorphism group of the unit co-disk bundle of any closed manifold. A similar conclusion holds for the ${\rm Ham}$-orbit space of an admissible Lagrangian in any Liouville domain. Moreover, we show that if a closed symplectic manifold contains an incompressible Lagrangian with a certain topological condition, then its Hamiltonian diffeomorphism group admits infinite-dimensional flats. Proofs of all these results rely on the existence of a family of heavy hypersurfaces.

Spectrally-large scale geometry via set-heaviness

TL;DR

The paper investigates when the spectral norm on the Hamiltonian diffeomorphism group of a Liouville domain yields unbounded large-scale geometry, linking this to the non-vanishing of symplectic cohomology via heaviness and heavy hypersurfaces. It develops a framework unifying Liouville-domain and closed-manifold settings, showing that implies the existence of rank- quasi-flats in , and extends the theory to Lagrangian orbit spaces using wrapped Floer cohomology with -heavy notions. The work outlines two main strategies—boundary-depth-based embeddings and constructions from heavy subsets—along with alternative approaches using egg-beater models and homogenized quasi-morphisms, yielding a robust dichotomy: either the spectral-norm geometry is bounded or it contains rank- flats. These results apply to unit co-disk bundles and other Liouville domains, revealing deep connections between symplectic cohomology, heaviness, and large-scale geometry in both absolute and relative (Lagrangian) settings.

Abstract

We show that there exist infinite-dimensional quasi-flats in the compactly supported Hamiltonian diffeomorphism group of the Liouville domain, with respect to the spectral norm, if and only if the symplectic cohomology of this Liouville domain does not vanish. In particular, there exist infinite-dimensional quasi-flats in the compactly supported Hamiltonian diffeomorphism group of the unit co-disk bundle of any closed manifold. A similar conclusion holds for the -orbit space of an admissible Lagrangian in any Liouville domain. Moreover, we show that if a closed symplectic manifold contains an incompressible Lagrangian with a certain topological condition, then its Hamiltonian diffeomorphism group admits infinite-dimensional flats. Proofs of all these results rely on the existence of a family of heavy hypersurfaces.

Paper Structure

This paper contains 20 sections, 35 theorems, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Liouville domain
  3. Admissible Lagrangian
  4. Closed manifold
  5. Summary of current art
  6. Coefficients
  7. Hofer norm in Lagrangian setting
  8. Hamiltonian Floer theory and spectral invariant
  9. Liouville domains
  10. Admissible Lagrangians
  11. Closed manifolds
  12. Proofs
  13. Proof of Proposition \ref{['thm-partial-heavy']}
  14. Proof of Lemma \ref{['lem-flat-1']}
  15. Proof of Proposition \ref{['prop-displace']}
  16. ...and 5 more sections

Key Result

Theorem A

Let $(W,\omega)$ be a Liouville domain, then $\mathop{\mathrm{SH}}\nolimits^*(W,\omega)\neq 0$ if and only if there exists a quasi-isometric embedding from $(C_{c}^\infty(I), d_{\infty})$ to $(\mathop{\mathrm{Ham}}\nolimits(W,\omega), d_{\gamma})$, where $(C_{c}^\infty(I), d_{\infty})$ is the metric

Figures (3)

  • Figure 1: The function $f_a$ associated to $a=(a_i)_{i=1}^\infty\in\mathbb{R}^\infty$.
  • Figure 2: $\widetilde{f}^{-1}(\max_{s\in[0,1]}f)$ and $\widetilde{f}^{-1}(\min_{s\in[0,1]}f)$ are heavy
  • Figure 3: Hamiltonian functions $F, H$ and $F_0$ in the collar region.

Theorems & Definitions (74)

  • Remark 1.1
  • Theorem A
  • Proposition 1.2
  • proof
  • Example 1.3
  • Proposition 1.4
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7: cf. Proposition D in Mai24
  • Example 1.8
  • ...and 64 more