On the existence of disk-like global sections for Reeb flows on the tight 3-sphere.

*(English)*Zbl 1246.53114Let \(\lambda\) be a contact form on \(S^3\). A disc-like global surface of section for the Reeb vector field \(R\) associated to \(\lambda\) is an embedded disk \(D\hookrightarrow S^3\) such that \(\partial D\) is a closed Reeb orbit, \(R\) is transverse to \(\mathrm{int}(D)\), and all Reeb trajectories in \(S^3\setminus \partial D\) hit \(\mathrm{int}(D)\) infinitely often forward and backward in time.

The following interesting problem is considered: given a tight contact form \(\lambda\) (the corresponding contact structure \(\xi=\ker\lambda\) is tight), which closed Reeb orbits are the boundary of some disc-like global surface of section? The existence of global sections of Reeb flows is motivated by the result of H. Hofer, K. Wysocki and E. Zehnder, who proved that a strictly contact 3-manifold \((M,\xi)\) is contactomorphic to a tight 3-sphere \(S^3\) if and only if \(\xi=\ker\lambda\) for some contact form \(\lambda\) whose Reeb vector field \(R\) has a nondegenerate periodic orbit which spans an embedded disk transversal to \(R\) and whose Conley-Zehnder index is 3 [Duke Math. J. 81, No. 1, 159–226 (1995; Zbl 0861.57026); correction ibid. 89, No. 3, 603–617 (1997; Zbl 0903.57009)].

The main result of this paper is the following characterization:

Let \(P'=(x',T')\) be a simply covered closed Reeb orbit associated to a nondegenerate tight form \(\lambda\) on \(S^3\). Then \(P'\) bounds a disc-like global surface of section for the Reeb flow if and only if \(P'\) is unknotted, its Conley-Zehnder index is greater than or equal to 3, \(P'\) has self-linking number \(-1\), and all periodic orbits \(P\) with Conley-Zehnder index equal 2 are linked to \(P'\).

Here, a periodic Reeb orbit \(P\) is a pair \((x,T)\) in which \(x\) is a Reeb trajectory and \(T>0\) is a period of \(x\). It is simply covered if \(T\) is its prime period. \(P=(x,T)\) is said to be linked to the unknotted orbit \(P'=(x',T')\) if the homology class of \(t\in \mathbb R/\mathbb Z\mapsto x(Tt)\) in \(H_1(S^3\setminus x'(\mathbb R),\mathbb Z)\) is nonzero. A periodic Reeb orbit \(P=(x,T)\) is called nondegenerate if 1 is not an eigenvalue of the linearized time \(T\)-map: \(\xi_{x(0)}\to \xi_{x(T)}=\xi_{x(0)}\); \(\lambda\) is called nondegenerate if every periodic orbit is nondegenerate.

The following interesting problem is considered: given a tight contact form \(\lambda\) (the corresponding contact structure \(\xi=\ker\lambda\) is tight), which closed Reeb orbits are the boundary of some disc-like global surface of section? The existence of global sections of Reeb flows is motivated by the result of H. Hofer, K. Wysocki and E. Zehnder, who proved that a strictly contact 3-manifold \((M,\xi)\) is contactomorphic to a tight 3-sphere \(S^3\) if and only if \(\xi=\ker\lambda\) for some contact form \(\lambda\) whose Reeb vector field \(R\) has a nondegenerate periodic orbit which spans an embedded disk transversal to \(R\) and whose Conley-Zehnder index is 3 [Duke Math. J. 81, No. 1, 159–226 (1995; Zbl 0861.57026); correction ibid. 89, No. 3, 603–617 (1997; Zbl 0903.57009)].

The main result of this paper is the following characterization:

Let \(P'=(x',T')\) be a simply covered closed Reeb orbit associated to a nondegenerate tight form \(\lambda\) on \(S^3\). Then \(P'\) bounds a disc-like global surface of section for the Reeb flow if and only if \(P'\) is unknotted, its Conley-Zehnder index is greater than or equal to 3, \(P'\) has self-linking number \(-1\), and all periodic orbits \(P\) with Conley-Zehnder index equal 2 are linked to \(P'\).

Here, a periodic Reeb orbit \(P\) is a pair \((x,T)\) in which \(x\) is a Reeb trajectory and \(T>0\) is a period of \(x\). It is simply covered if \(T\) is its prime period. \(P=(x,T)\) is said to be linked to the unknotted orbit \(P'=(x',T')\) if the homology class of \(t\in \mathbb R/\mathbb Z\mapsto x(Tt)\) in \(H_1(S^3\setminus x'(\mathbb R),\mathbb Z)\) is nonzero. A periodic Reeb orbit \(P=(x,T)\) is called nondegenerate if 1 is not an eigenvalue of the linearized time \(T\)-map: \(\xi_{x(0)}\to \xi_{x(T)}=\xi_{x(0)}\); \(\lambda\) is called nondegenerate if every periodic orbit is nondegenerate.

Reviewer: Božidar Jovanović (Beograd)

##### MSC:

53D35 | Global theory of symplectic and contact manifolds |

57R17 | Symplectic and contact topology in high or arbitrary dimension |

37J55 | Contact systems |

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\textit{U. Hryniewicz} and \textit{P. A. S. Salomão}, Duke Math. J. 160, No. 3, 415--465 (2011; Zbl 1246.53114)

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