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Flexibility of the Hamiltonian adjoint action and classification of bi-invariant metrics

Lev Buhovsky, Maksim Stokić

TL;DR

This work addresses the geometry of bi-invariant metrics on the Hamiltonian diffeomorphism group of connected symplectic manifolds, with a focus on open, exact cases. The authors prove a strong local-to-global flexibility of the Hamiltonian adjoint action: any compactly supported function with bounded sup-norm can be expressed as a finite combination of pullbacks of a fixed non-constant function, with coefficients controlled by the support volume, leading to universal domination of invariant norms by $\|\cdot\|_{\infty}+\|\cdot\|_{L^1}$. Using this, they classify all intrinsic bi-invariant pseudo-metrics on $\mathrm{Ham}(M,\omega)$ for exact symplectic manifolds: a degenerate Calabi-type metric $\mu|\mathrm{Cal}|$ or a non-degenerate metric equivalent (up to constants) to Hofer’s metric, or Hofer+$|\mathrm{Cal}|$, thus resolving a question of Eliashberg–Polterovich. The results unify Hofer geometry for open manifolds and provide a precise framework linking Calabi, Hofer, and adjoint-action flexibility in the exact-symplectic setting.

Abstract

On an open, connected symplectic manifold $(M,ω)$, the group of Hamiltonian diffeomorphisms forms an infinite-dimensional Fréchet Lie group with Lie algebra $C^{\infty}_c(M)$ and adjoint action given by pullbacks. We prove that this action is flexible: for any non-constant $u \in C^{\infty}(M)$, every $f \in C^{\infty}_c(M)$ can be expressed as a weighted finite sum of elements from the adjoint orbit of $u$, with total weight bounded by constant multiple of $\|f\|_{\infty} + \|f\|_{L^1}$. Consequently, all $\mathrm{Ham}(M,ω)$-invariant norms on $C^{\infty}_c(M)$ are dominated by a sum of $L^{\infty}$ and $L^1$ norms. As an application, we classify up to equivalence all bi-invariant pseudo-metrics on the group of Hamiltonian diffeomorphisms of an exact symplectic manifold, answering a question of Eliashberg and Polterovich.

Flexibility of the Hamiltonian adjoint action and classification of bi-invariant metrics

TL;DR

This work addresses the geometry of bi-invariant metrics on the Hamiltonian diffeomorphism group of connected symplectic manifolds, with a focus on open, exact cases. The authors prove a strong local-to-global flexibility of the Hamiltonian adjoint action: any compactly supported function with bounded sup-norm can be expressed as a finite combination of pullbacks of a fixed non-constant function, with coefficients controlled by the support volume, leading to universal domination of invariant norms by . Using this, they classify all intrinsic bi-invariant pseudo-metrics on for exact symplectic manifolds: a degenerate Calabi-type metric or a non-degenerate metric equivalent (up to constants) to Hofer’s metric, or Hofer+, thus resolving a question of Eliashberg–Polterovich. The results unify Hofer geometry for open manifolds and provide a precise framework linking Calabi, Hofer, and adjoint-action flexibility in the exact-symplectic setting.

Abstract

On an open, connected symplectic manifold , the group of Hamiltonian diffeomorphisms forms an infinite-dimensional Fréchet Lie group with Lie algebra and adjoint action given by pullbacks. We prove that this action is flexible: for any non-constant , every can be expressed as a weighted finite sum of elements from the adjoint orbit of , with total weight bounded by constant multiple of . Consequently, all -invariant norms on are dominated by a sum of and norms. As an application, we classify up to equivalence all bi-invariant pseudo-metrics on the group of Hamiltonian diffeomorphisms of an exact symplectic manifold, answering a question of Eliashberg and Polterovich.

Paper Structure

This paper contains 13 sections, 17 theorems, 133 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Bi-invariant metrics on $\mathrm{Ham}(M,\omega)$
  3. Exact symplectic manifolds and Calabi homomorphism
  4. Proof of Theorem \ref{['AuxilaryTheorem']}
  5. Proof of Theorem \ref{['MainTheorem']}
  6. Proof of Proposition \ref{['PropositionL1-Splitting']}
  7. Proof of Theorem \ref{['Theorem:Upper_Bound_Infinite_volume']}
  8. Proof of Proposition \ref{['Prop:Approximation_by_Step_function']}
  9. Proof of Claim \ref{['Claim:Sikorav_trick']}
  10. Proof of Claim \ref{['Claim:Setup_for_Sikorav_trick']}
  11. Proof of Theorem \ref{['Theorem:Classification_of_bi-invariant_metrics']}
  12. Appendix: Proof of Theorem \ref{['Theorem:Lower_Bound_Open_Case']}
  13. Proof of Theorem \ref{['Theorem:Lower_Bound_Open_Case']}

Key Result

Theorem 1

Let $(M,\omega)$ be an open and connected symplectic manifold, and let $u\in C^{\infty}(M)$ be a non-constant function. There exists a constant $c=c(u)>0$ such that for any $f\in C^{\infty}_{0,c}(M)$ with $\|f\|_{\infty}\leq 1$ one can write for some Hamiltonian diffeomorphisms $\Phi_{i,\pm}\in\mathrm{Ham}_c(M,\omega)$.

Figures (1)

  • Figure 1: Cubes $\widetilde{Q}_i$ subdivided into smaller cubes

Theorems & Definitions (46)

  • Theorem 1
  • Corollary 1.1
  • Theorem 2
  • Corollary 1.2
  • Theorem 3
  • Theorem 4
  • Theorem 5: Classification of bi-invariant pseudo-metrics on $\mathrm{Ham}(M,\omega)$
  • Claim 2.1
  • proof
  • Proposition 2.2
  • ...and 36 more