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Cylindrical contact homology for weakly convex contact forms in dimension three

Ana Kelly de Oliveira, Pedro A. S. Salomão

Abstract

A contact form $λ$ on a closed contact three-manifold $(M,ξ)$ is called weakly convex if either it has no contractible Reeb orbit, or the first Chern class of $ξ$ vanishes on $π_2(M)$, and the index of every contractible Reeb orbit is at least $2$. We present conditions for a weakly convex contact form to admit a well-defined cylindrical contact homology. The key point is a cancellation mechanism for boundary degenerations involving index-2 Reeb orbits, based on a parity property of holomorphic planes.

Cylindrical contact homology for weakly convex contact forms in dimension three

Abstract

A contact form on a closed contact three-manifold is called weakly convex if either it has no contractible Reeb orbit, or the first Chern class of vanishes on , and the index of every contractible Reeb orbit is at least . We present conditions for a weakly convex contact form to admit a well-defined cylindrical contact homology. The key point is a cancellation mechanism for boundary degenerations involving index-2 Reeb orbits, based on a parity property of holomorphic planes.

Paper Structure

This paper contains 15 sections, 39 theorems, 111 equations, 5 figures.

Table of Contents

  1. Introduction and main results
  2. Basics
  3. Reeb flows
  4. Pseudo-holomorphic curves
  5. The Conley-Zehnder index
  6. The asymptotic operator
  7. The linearization of the Cauchy-Riemann Operator
  8. Holomorphic Buildings
  9. Buildings of index 1 and 2
  10. Ruling out buildings as in Theorem \ref{['C']}-(iii.4) and (iii.5)
  11. Buildings as in Theorem \ref{['C']}-(iii.6)
  12. Gluing of buildings
  13. Coherent orientations
  14. The existence of the opposite plane
  15. Proof of main results

Key Result

Theorem 1.1

Let $\lambda$ be a nondegenerate contact form on a closed contact three-manifold $($M$,\xi)$. Assume that the Reeb flow of $\lambda$ either has no contractible Reeb orbit, or the first Chern class $c_1(\xi)$ vanishes on $\pi_2(M)$ and the index of every contractible Reeb orbit is at least $3$. Assum

Figures (5)

  • Figure 1: $J$-holomorphic buildings in Theorem \ref{['C']}-(iii)
  • Figure 2: Holomorphic buildings of Cases 1 and 2.
  • Figure 3: Braids $\zeta^k_{1,2}$ and $\zeta^k_{\pm}$ in the image of the curves $w_k$ that converge to the building $u$.
  • Figure 4: J-holomorphic building $\mathcal{B}= (v_1,v_2)$
  • Figure 5: The buildings $\mathcal{B}$ and $\mathcal{B}'$ at the boundary of the compactification of $\mathcal{M}^J_{0,2}(\alpha_1^d;\alpha_2^d)/\mathbb{R}$ have opposite signs.

Theorems & Definitions (59)

  • Theorem 1.1: Hutchings-Nelson HN
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1: Hofer Hofer1993prop1
  • Theorem 2.2: Hofer-Wysocki-Zehnder prop2
  • Theorem 2.3: Hofer-Wysocki-Zehnder prop1
  • Proposition 2.4
  • Proposition 2.5
  • Definition 2.6
  • Lemma 2.7: prop2
  • ...and 49 more