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Periodic orbits of non-degenerate lacunary contact forms on prequantization bundles

Miguel Abreu, Leonardo Macarini

Abstract

A non-degenerate contact form is lacunary if the indexes of every contractible periodic Reeb orbit have the same parity. To the best of our knowledge, every contact form with finitely many periodic orbits known so far is non-degenerate and lacunary. We show that every non-degenerate lacunary contact form on a suitable prequantization of a closed symplectic manifold $B$ has precisely $r_B$ contractible closed orbits, where $r_B=\dim \mathrm{H}_*(B;{\mathbb Q})$. Examples of such prequantizations include the standard contact sphere, the unit cosphere bundle of a compact rank one symmetric space (CROSS) and many others. We also consider some prequantizations of orbifolds, like lens spaces and the unit cosphere bundle of lens spaces, and obtain multiplicity results for these prequantizations.

Periodic orbits of non-degenerate lacunary contact forms on prequantization bundles

Abstract

A non-degenerate contact form is lacunary if the indexes of every contractible periodic Reeb orbit have the same parity. To the best of our knowledge, every contact form with finitely many periodic orbits known so far is non-degenerate and lacunary. We show that every non-degenerate lacunary contact form on a suitable prequantization of a closed symplectic manifold has precisely contractible closed orbits, where . Examples of such prequantizations include the standard contact sphere, the unit cosphere bundle of a compact rank one symmetric space (CROSS) and many others. We also consider some prequantizations of orbifolds, like lens spaces and the unit cosphere bundle of lens spaces, and obtain multiplicity results for these prequantizations.
Paper Structure (16 sections, 17 theorems, 94 equations, 1 table)

This paper contains 16 sections, 17 theorems, 94 equations, 1 table.

Table of Contents

  1. Introduction
  2. Introduction and Main Results
  3. Organization of the paper
  4. Acknowledgments
  5. Equivariant symplectic homology
  6. Equivariant symplectic homology
  7. Equivariant symplectic homology of prequantizations
  8. Resonance relations
  9. Index recurrence
  10. Proof of Theorem \ref{['thm:main']} under the assumption that $c_B>n/2$
  11. Outline of the proof
  12. Proof of the theorem
  13. Proof of Theorem \ref{['thm:main orbifolds']}
  14. Multiplicity of periodic orbits for lens spaces and their unit cosphere bundles
  15. Final questions
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $(M^{2n+1},\xi)$ be a prequantization $S^1$-bundle of a closed symplectic manifold $(B,\omega)$ such that $\omega|_{\pi_2(B)}\neq 0$, $c_B>n/2$ and ${\mathrm{H}}_{k}(B;\mathbb{Q})=0$ for every odd $k$. Let $\alpha$ be a non-degenerate contact form supporting $\xi$ which is index-positive and has Then $\alpha$ carries at least $r_B$ geometrically distinct contractible periodic orbits, where $r_

Theorems & Definitions (44)

  • Theorem 1.1: GGM2
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Corollary 1.5: GGM2
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 34 more