##
**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.

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.

Reviewer: Antonio Lerario (Triest)

### MSC:

53D35 | Global theory of symplectic and contact manifolds |

35J62 | Quasilinear elliptic equations |

53D10 | Contact manifolds (general theory) |

32Q65 | Pseudoholomorphic curves |

### References:

[1] | C. Abbas, Pseudoholomorphic strips in symplectizations, II: Fredholm theory and transversality , Comm. Pure Appl. Math. 57 (2004), 1-58. · Zbl 1073.53104 |

[2] | -, Pseudoholomorphic strips in symplectizations, III: Embedding properties and compactness , J. Symplectic Geom. 2 (2004), 219-260. · Zbl 1085.53072 |

[3] | -, Introduction to compactness results in symplectic field theory , in preparation. |

[4] | C. Abbas, K. Cieliebak, and H. Hofer, The Weinstein conjecture for planar contact structures in dimension three , Comment. Math. Helv. 80 (2005), 771-793. · Zbl 1098.53063 |

[5] | C. Abbas and H. Hofer, Holomorphic curves and global questions in contact geometry , in preparation. |

[6] | C. Abbas, H. Hofer, and S. Lisi, Renormalization and energy quantization in Reeb dynamics , in preparation. |

[7] | -, Some applications of a homological perturbed Cauchy-Riemann equation , in preparation. |

[8] | L. V. Ahlfors, Lectures on Quasiconformal Mappings , Van Nostrand Math. Stud. 10 , D. Van Nostrand, Toronto, 1966. · Zbl 0138.06002 |

[9] | L. V. Ahlfors and L. Bers, Riemann’s mapping theorem for variable metrics , Ann. of Math. (2) 72 (1960), 385-404. JSTOR: · Zbl 0104.29902 |

[10] | J. W. Alexander, A lemma on systems of knotted curves , Proc. Natl. Acad. Sci. USA 9 (1923), 93-95. · JFM 49.0408.03 |

[11] | L. Bers, Riemann Surfaces , lectures, New York University, 1957-1958. |

[12] | L. Bers and L. Nirenberg, “On a representation theorem for linear elliptic systems with discontinuous coefficients and its applications” in Convegno internazionale sulle equazioni lineari alle derivate parziali, Trieste (1954) , 111-140, Cremonese, Rome, 1955. · Zbl 0067.32503 |

[13] | S.-S. Chern, An elementary proof of the existence of isothermal parameters on a surface , Proc. Amer. Math. Soc. 6 (1955), 771-782. JSTOR: · Zbl 0066.15402 |

[14] | J. B. Etnyre, Planar open book decompositions and contact structures , Int. Math. Res. Not. IMRN 2004 , no. 79, 4255-4267. · Zbl 1069.57016 |

[15] | O. Forster, Lectures on Riemann Surfaces , Grad. Texts in Math. 81 , Springer, New York, 1981. · Zbl 0475.30002 |

[16] | E. Giroux, “Géométrie de contact: De la dimension trois vers les dimensions supérieures” in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) , Higher Ed. Press, Beijing, 2002, 405-414. · Zbl 1015.53049 |

[17] | H. Hofer, Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three , Invent. Math. 114 (1993), 515-563. · Zbl 0797.58023 |

[18] | -, “Holomorphic curves and real three-dimensional dynamics” in GAFA 2000 (Tel Aviv, 1999) , Geom. Funct. Anal. 2000 , Special Volume, Part II, 674-704. |

[19] | H. Hofer, K. Wysocki, and E. Zehnder, Properties of pseudoholomorphic curves in symplectizations, II: Embedding controls and algebraic invariants , Geom. Funct. Anal. 5 (1995), 270-328. · Zbl 0845.57027 |

[20] | -, The dynamics on three-dimensional strictly convex energy surfaces , Ann. of Math. (2) 148 (1998), 197-289. JSTOR: · Zbl 0944.37031 |

[21] | -, Properties of pseudoholomorphic curves in symplectizations,. III: Fredholm theory , Progr. Nonlinear Differential Equations Appl. 35 , Birkhäuser, Basel, 1999, 381-475. · Zbl 0924.58003 |

[22] | -, Finite energy foliations of tight three-spheres and Hamiltonian dynamics , Ann. of Math. (2) 157 (2003), 125-255. JSTOR: · Zbl 1215.53076 |

[23] | H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics , Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser, Basel, 1994. |

[24] | C. Hummel, Gromov’s Compactness Theorem for Pseudoholomorphic Curves , Prog. Math. 151 , Birkhäuser, Basel, 1997. · Zbl 0870.53002 |

[25] | P. D. Lax, Functional Analysis , John Wiley, New York, 2001. · Zbl 1009.47001 |

[26] | L. Lichtenstein, Zur Theorie der konformen Abbildung nichtanalytischer, singularitätenfreier Flächenstücke auf ebene Gebiete , J. Krak. Auz. (1916), 192-217. · JFM 46.0547.01 |

[27] | D. Mcduff and D. Salamon, J-holomorphic Curves and Symplectic Topology , Amer. Math. Soc. Colloq. Publ. 52 , Amer. Math. Soc., Providence, 2004. · Zbl 1064.53051 |

[28] | M. J. Micallef and B. White, The structure of branch points in minimal surfaces and in pseudoholomorphic curves , Ann. of Math. (2) 141 (1995), 35-85. JSTOR: · Zbl 0873.53038 |

[29] | C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations , Trans. Amer. Math. Soc. 43 , no. 1 (1938), 126-166. JSTOR: · Zbl 0018.40501 |

[30] | D. Rolfsen, Knots and Links , Mathematics Lecture Series 7 , Publish or Perish, Berkeley, Cali., 1976. · Zbl 0339.55004 |

[31] | R. Siefring, Relative asymptotic behavior of pseudoholomorphic half-cylinders , Comm. Pure Appl. Math. 61 (2008), 1631-1684. · Zbl 1158.53068 |

[32] | E. M. Stein, Singular Integrals and Differentiability Properties of Functions , Princeton Math. Ser. 30 , Princeton Univ. Press, Princeton, 1970. · Zbl 0207.13501 |

[33] | C. H. Taubes, The Seiberg-Witten equations and the Weinstein conjecture , Geom. Topol. 11 (2007), 2117-2202. · Zbl 1135.57015 |

[34] | -, The Seiberg-Witten equations and the Weinstein conjecture, II: More closed integral curves of the Reeb vector field , Geom. Topol. 13 (2009), 1337-1417. · Zbl 1200.57018 |

[35] | W. P. Thurston and H. E. Winkelnkemper, On the existence of contact forms , Proc. Amer. Math. Soc. 52 (1975), 345-347. JSTOR: · Zbl 0312.53028 |

[36] | A. J. Tromba, Teichmüller Theory in Riemannian Geometry , Lectures Math. ETH Zürich, Birkhäuser, Basel, 1992. |

[37] | A. Weinstein, On the hypotheses of Rabinowitz’ periodic orbit theorems , J. Differential Equations 33 (1979), 353-358. · Zbl 0388.58020 |

[38] | C. Wendl, Finite energy foliations on overtwisted contact manifolds , Geom. Topol. 12 (2008) 531-616. · Zbl 1141.53082 |

[39] | -, Open book decompositions and stable Hamiltonian structures , Expo. Math. 28 (2010), 187-197. · Zbl 1198.32012 |

[40] | -, Strongly fillable contact manifolds and J-holomorphic foliations , Duke Math. J. 151 (2010), 337-387. · Zbl 1207.32022 |

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