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On the existence of disk-like global sections for Reeb flows on the tight 3-sphere. (English) Zbl 1246.53114
Let $$\lambda$$ 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.

##### MSC:
 53D35 Global theory of symplectic and contact manifolds 57R17 Symplectic and contact topology in high or arbitrary dimension 37J55 Contact systems
##### Citations:
Zbl 0861.57026; Zbl 0903.57009
Full Text:
##### References:
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