Holomorphic open book decompositions. (English) Zbl 1230.53077

Every closed 3-dimensional orientable manifold admits an open book decomposition, i.e., there exists a link \(K\) in \(M\) and a fibration \(\tau: M\backslash K \to S^1\) whose fibers are interiors of compact embedded surfaces with common boundary \(K\). If \(M\) is endowed with a contact form \(\lambda\), it is said that the open book decomposition supports \(\lambda\) if there exists a contact form \(\eta\) with \(\ker \lambda=\ker \eta\) and such that \(d\eta\) induces an area form on each fiber of \(\tau\), with \(K\) consisting of closed orbits of the Reeb vector field of \(\eta\) and oriented by \(\eta\) as the boundary of each fiber. The existence of an open book decomposition supporting a co-oriented contact structure was proved by E. Giroux [“Géométrie de contact: de la dimension trois vers les dimensions supérieures”, in: T. T. Li (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20–28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 405–414 (2002; Zbl 1015.53049)].
In the current paper, the author shows that the supporting open book decomposition can be chosen in such a way that the fibers of \(\tau\) solve a homological perturbed Cauchy-Riemann equation. More precisely, if \(\pi: TM\to \ker \eta\) is the transversal projection along the Reeb vector field of \(\eta\) and \(J:\ker \eta \to \ker \eta\) is an almost complex structure such that \(d\eta(\cdot,J\cdot)\) defines a positive definite form on the fibers of \(\tau\), a nonlinear elliptic system of first order is considered. Its solutions, in the author’s notation, are 5-tuples \((S,j,\Gamma, \tilde u, \gamma)\) where \(S\) is a Riemann surface with complex structure \(j,\) \(\Gamma\subset S\) is a finite set, \(\tilde u=(a, u):S\backslash \Gamma\to \mathbb{R}\times M\) is a proper map and \(\gamma\) is a one-form on \(S\) such that (i) \(u|_{S\backslash \Gamma}\) is a \(J\)-holomorphic curve; (ii) \((u^*\eta)\circ j=da+\gamma\) on \(S\backslash \Gamma\); (iii) \(d\gamma=d(\gamma \circ j)=0\) on \(S\); (iv) \(\tilde u\) satisfies a finite energy condition. The main result is thus the existence – for the contact closed 3-manifold \((M, \lambda)\) – of a smooth family of solutions \((S, j_\theta, \Gamma_\theta, \tilde{u}_\theta=(a_\theta, u_\theta), \gamma_\theta)_{\theta\in S^1}\) of the previous system for a suitable \(J\) and \(\eta=f\lambda\), where \(f\) is a positive function on \(M\) such that these solutions give an open book decomposition of \(M\) supporting \(\lambda\) (the binding \(K\) is the common asymptotic limit of all maps \(u_\theta\) at the punctures).
The motivation for such a study, as noticed by the author, is the fact that the Weinstein conjecture for \((M,\eta)\), i.e., the existence of a closed trajectory of the Reeb vector field of \(\eta\), is equivalent to the existence of a non-constant solution to the previous system. On the other hand, the author himself proved the Weinstein conjecture for planar contact structures using this idea [the author, K. Cieliebak and H. Hofer, “The Weinstein conjecture for planar contact structures in dimension three”, Comment. Math. Helv. 80, No. 4, 771–793 (2005; Zbl 1098.53063)].
The strategy for the proof of the main theorem is to start with a supporting open book decomposition as in [Giroux, loc. cit.] and to deform it to the desired one – first locally and then globally – using a compactness result for equations of perturbed holomorphic curves.


53D35 Global theory of symplectic and contact manifolds
35J62 Quasilinear elliptic equations
53D10 Contact manifolds (general theory)
32Q65 Pseudoholomorphic curves
Full Text: DOI arXiv


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