Real holomorphic curves and invariant global surfaces of section (Q2800461)

The main result of this paper (Theorem 2.3) asserts that if the boundary of an invariant star-shaped domain is in addition dynamically convex, then it carries an invariant global disk-like surface of section.NEWLINENEWLINENEWLINEThe authors say that the topic of this paper is motivated by recent progress of applying methods from holomorphic curve theory to the study of the dynamics of the restricted three-body problem [\textit{P. Albers} et al., Arch. Ration. Mech. Anal. 204, No. 1, 273--284 (2012; Zbl 1287.70006)]. The Hamiltonian of the restricted three-body problem is invariant under an antisymplectic involution [\textit{H. Poincaré}, New methods in celestial mechanics. 1. Periodic and asymptotic solutions. 2. Approximations by series. 3. Integral invariants and asymptotic properties of certain solutions. Ed. and introduced by Daniel Goroff. Transl. from the French orig. 1892--1899. Bristol: American Institute of Physics (1993; Zbl 0776.01009)]. This fact is translated into the fact that the star-shaped domains are invariant under complex conjugation.NEWLINENEWLINENEWLINETo produce a global disk-like surface of section via a stretching argument, techniques from holomorphic curve theory combined with methods from Symplectic Field theory are applied in [\textit{H. Hofer} et al., Ann. Math. (2) 157, No. 1, 125--255 (2003; Zbl 1215.53076)]. Hereafter referred to as [\textit{D. L. Dragnev}, Commun. Pure Appl. Math. 57, No. 6, 726--763 (2004; Zbl 1063.53086)]. This method is applied in this paper, but uses real holomorphic curve theory which means to apply techniques from open string theory instead of closed string theory.NEWLINENEWLINENEWLINEAfter defining the notions of finite energy plane and invariant global surface of section (Definitions 2.1 and 2.2), \(M\subset \mathbb C^2\) has an invariant global disk-like surface of section \(\mathcal{D}\) for the Reeb flow such that the Conley-Zehnder index of the spanning orbit of \(\mathcal{D}\) is either 3 or 4, and has some additional properties; if it is a star-shaped hypersurface invariant under the antisymplectic involution \(\tilde{\rho}\) such that \((M,X)\) in \(\mathbb C^2\) is dynamically convex (Theorem 2.3).NEWLINENEWLINENEWLINEThe main part of the proof of Theorem 2.3 is the study of the asymptotic behavior of a finite energy surface, which is related to the Conley-Zehnder indices of the asymptotic periodic orbit. This is done by establishing a relation between the Conley-Zehnder index and the Robbins-Salamon index (Proposition 3.6) via winding number expressions of these indices (described in \S3.2). Then it is shown that the Robbins-Salamon index of a Reeb chord is a half integer, and the Conley-Zehnder index of a periodic Reeb orbit is an integer.NEWLINENEWLINENEWLINEProperties of invariant finite energy spheres are studied in \S4. Especially, introducing the notion of ``somewhere injective'' for finite energy half-spaces (Definition 4.11), the existence of a set of punctures with suitable properties for a not somewhere injective invariant finite energy sphere is shown (Theorem 4.13).NEWLINENEWLINENEWLINESection 5 improves some results on real holomorphic curves in \(\mathbb{CP}^2\), and \(\mathbb{RP}^2\) given in [loc. cit.]. After these preparations, Theorem 2.3 is proved in \S6.