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Birkhoff sections in 3-manifold with invariant toric foliation

Wentian Kuang

TL;DR

The paper develops a constructive framework to study Birkhoff sections for flows on 3-manifolds that admit a regular invariant toric foliation, providing necessary and sufficient conditions for boundary orbits and a consistent definition of transverse curves across invariant tori. It shows how boundary and broken torus data constrain the homology type of transverse curves, enabling explicit construction of Birkhoff sections and, in favorable cases, disk-like global surfaces; the approach yields compute-able genus and Euler characteristics. The results are applied to boundaries of toric domains and to energy hypersurfaces of separable Hamiltonians, delivering concrete criteria for disk-like or annulus-like sections, and offering an alternative route to known dynamical-convexity and capacity results. Overall, the work bridges toric foliation dynamics with global surface-of-section theory, providing explicit, verifiable conditions to produce and classify Birkhoff sections in integrable settings and connecting to broader Reeb-flow and symplectic-geometry contexts.

Abstract

In this paper, we study the Birkhoff sections in a 3-manifold foliated by invariant tori. We establish the necessary and sufficient conditions for various types of periodic orbits to serve as boundary orbits of a Birkhoff section. The construction relies on the dynamical behaviour of the flow combined with fundamental topological argument. As an application, we study the boundaries of toric domains and the energy hypersurfaces of separable Hamiltonian systems, providing conditions for the existence or non-existence of different types of Birkhoff sections. Additionally, we offer an alternative proof of part of the results presented in [23] and [14].

Birkhoff sections in 3-manifold with invariant toric foliation

TL;DR

The paper develops a constructive framework to study Birkhoff sections for flows on 3-manifolds that admit a regular invariant toric foliation, providing necessary and sufficient conditions for boundary orbits and a consistent definition of transverse curves across invariant tori. It shows how boundary and broken torus data constrain the homology type of transverse curves, enabling explicit construction of Birkhoff sections and, in favorable cases, disk-like global surfaces; the approach yields compute-able genus and Euler characteristics. The results are applied to boundaries of toric domains and to energy hypersurfaces of separable Hamiltonians, delivering concrete criteria for disk-like or annulus-like sections, and offering an alternative route to known dynamical-convexity and capacity results. Overall, the work bridges toric foliation dynamics with global surface-of-section theory, providing explicit, verifiable conditions to produce and classify Birkhoff sections in integrable settings and connecting to broader Reeb-flow and symplectic-geometry contexts.

Abstract

In this paper, we study the Birkhoff sections in a 3-manifold foliated by invariant tori. We establish the necessary and sufficient conditions for various types of periodic orbits to serve as boundary orbits of a Birkhoff section. The construction relies on the dynamical behaviour of the flow combined with fundamental topological argument. As an application, we study the boundaries of toric domains and the energy hypersurfaces of separable Hamiltonian systems, providing conditions for the existence or non-existence of different types of Birkhoff sections. Additionally, we offer an alternative proof of part of the results presented in [23] and [14].

Paper Structure

This paper contains 12 sections, 40 theorems, 23 equations, 4 figures.

Table of Contents

  1. Introduction
  2. History and some related results.
  3. Settings and main results
  4. Applications
  5. Boundaries of toric domains
  6. Separable Hamiltonian
  7. Transverse curves on a family of invariant tori
  8. Transverse curves near a transversal broken torus
  9. Transverse curves near the boundary orbits
  10. Applications
  11. Boundary of a toric domain
  12. Energy hypersurface of separable Hamiltonian

Key Result

Theorem 1.1

If $M$ admits a regular toric invariant foliation and the natural generators on different families of tori coincide, then there is a consistent way to define the homology of closed curves on invariant tori in $M$. Let $S$ be a Birkhoff section in $M$, we have where $(p_i^-,q_i^-),(p_j^+,q_j^+)$ are homology types of transverse curves on different families of tori in neighbourhood of $B$.

Figures (4)

  • Figure 4.1: Two types of transverse curves near a non-transversal invariant torus.
  • Figure 4.2: Fundamental domain for new rectangle coordinates with $P=3,Q=2$.
  • Figure 5.3: The limit curve on non-transversal torus, where the curve is illustrated in $\mathbb{R}^2$ and the torus is represented by $\mathbb{R}^2/\mathbb{Z}^2$.
  • Figure 5.4: Graph of $g(y)\le 1$.

Theorems & Definitions (71)

  • Definition 1
  • Definition 2
  • Theorem 1.1: Theorem \ref{['thm:main']}
  • Corollary 1.2
  • proof
  • Theorem 1.3: Theorem \ref{['thm:birktoric']}
  • Corollary 1.4
  • Remark 1
  • Theorem 1.5: [Theorem 1.1 in vanKoert2022]
  • Theorem 1.6: Theorem 1.7 in Hutchings2022
  • ...and 61 more