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Bott-integrable Reeb flows on 3-manifolds

Hansjörg Geiges, Jakob Hedicke, Murat Sağlam

Abstract

This paper is devoted to studying a notion of Bott integrability for Reeb flows on contact 3-manifolds. We show, in analogy with work of Fomenko-Zieschang on Hamiltonian flows in dimension 4, that Bott-integrable Reeb flows exist precisely on graph manifolds. We also show that all $S^1$-invariant contact structures on Seifert manifolds, as well as all contact structures on the 3-sphere, on the 3-torus, and on $S^1\times S^2$, admit Bott-integrable Reeb flows. Along the way, we establish some general Liouville-type theorems for Bott-integrable Reeb flows, and a number of topological constructions (connected sum, open books, Dehn surgery) that may be expected to have wider applications.

Bott-integrable Reeb flows on 3-manifolds

Abstract

This paper is devoted to studying a notion of Bott integrability for Reeb flows on contact 3-manifolds. We show, in analogy with work of Fomenko-Zieschang on Hamiltonian flows in dimension 4, that Bott-integrable Reeb flows exist precisely on graph manifolds. We also show that all -invariant contact structures on Seifert manifolds, as well as all contact structures on the 3-sphere, on the 3-torus, and on , admit Bott-integrable Reeb flows. Along the way, we establish some general Liouville-type theorems for Bott-integrable Reeb flows, and a number of topological constructions (connected sum, open books, Dehn surgery) that may be expected to have wider applications.
Paper Structure (49 sections, 22 theorems, 124 equations, 5 figures)

This paper contains 49 sections, 22 theorems, 124 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Bott integrability
  3. The main results
  4. Further non-existence statements
  5. The literature on contact integrability
  6. Bott integrability
  7. Motivating the definition
  8. Liouville theorems
  9. Creating isolated critical Reeb orbits
  10. $3$-manifolds admitting integrable Reeb flows
  11. Graph manifolds
  12. Integrable Reeb flows on the building blocks
  13. The solid torus
  14. Pair of pants times $S^1$
  15. Seifert fibred building blocks
  16. ...and 34 more sections

Key Result

Theorem 1.3

A closed, oriented $3$-manifold admits a Bott-integrable Reeb flow if and only if it is a graph manifold.

Figures (5)

  • Figure 1: Modifying the Lutz form into $\mathrm{d} x_2+r\,\mathrm{d} x_1$.
  • Figure 2: The modified Bott integral $f^*$.
  • Figure 3: Carrot pants.
  • Figure 4: Interpolating Lutz forms.
  • Figure 5: The model for the connected sum

Theorems & Definitions (50)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 1.6
  • Proposition 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Lemma 2.1
  • ...and 40 more