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Proof of Hofer-Wysocki-Zehnder's two or infinity conjecture

Dan Cristofaro-Gardiner, Umberto Hryniewicz, Michael Hutchings, Hui Liu

Abstract

We prove that every Reeb flow on a closed connected three-manifold has either two or infinitely many simple periodic orbits, assuming that the associated contact structure has torsion first Chern class. As a special case, we prove a conjecture of Hofer-Wysocki-Zehnder published in 2003 asserting that a smooth and autonomous Hamiltonian flow on $\mathbb{R}^4$ has either two or infinitely many simple periodic orbits on any regular compact connected energy level that is transverse to the radial vector field. Other corollaries settle some old problems about Finsler metrics: we show that every Finsler metric on $S^2$ has either two or infinitely many prime closed geodesics; and we show that a Finsler metric on $S^2$ with at least one closed geodesic that is not irrationally elliptic must have infinitely many prime closed geodesics. The novelty of our work is that we do not make any nondegeneracy hypotheses.

Proof of Hofer-Wysocki-Zehnder's two or infinity conjecture

Abstract

We prove that every Reeb flow on a closed connected three-manifold has either two or infinitely many simple periodic orbits, assuming that the associated contact structure has torsion first Chern class. As a special case, we prove a conjecture of Hofer-Wysocki-Zehnder published in 2003 asserting that a smooth and autonomous Hamiltonian flow on has either two or infinitely many simple periodic orbits on any regular compact connected energy level that is transverse to the radial vector field. Other corollaries settle some old problems about Finsler metrics: we show that every Finsler metric on has either two or infinitely many prime closed geodesics; and we show that a Finsler metric on with at least one closed geodesic that is not irrationally elliptic must have infinitely many prime closed geodesics. The novelty of our work is that we do not make any nondegeneracy hypotheses.
Paper Structure (41 sections, 53 theorems, 200 equations)

This paper contains 41 sections, 53 theorems, 200 equations.

Table of Contents

  1. Introduction
  2. The main result and corollaries
  3. Some history and related results
  4. Idea of the proof and organization of the paper
  5. Some open questions and a small step
  6. Preliminaries on embedded contact homology
  7. Orbit sets and the ECH index
  8. Holomorphic currents and the index inequality
  9. The definition of ECH
  10. The $U$ map
  11. Spectral invariants and the volume property
  12. Partition conditions
  13. The $J_0$ index and topological complexity bounds
  14. Criterion for a Birkhoff section
  15. Birkhoff sections for nondegenerate perturbations
  16. ...and 26 more sections

Key Result

Theorem 1.1

Let $Y$ be a closed connected three-manifold, let $\lambda$ be a contact form on $Y$, and let $\xi=\operatorname{Ker}(\lambda)$ be the associated contact structure. Assume that the first Chern class $c_1(\xi) \in H^2(Y;\mathbb{Z})$ is torsion. Then $\lambda$ has either two or infinitely many simple

Theorems & Definitions (123)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • proof
  • Remark 1.5
  • Remark 1.6
  • Corollary 1.7
  • proof
  • Corollary 1.8
  • ...and 113 more