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Pseudoholomorphic strips in symplectisations. III: Embedding properties and compactness. (English) Zbl 1085.53072

This is the last in the series of papers discussing behavior of pseudoholomorphic strips in the symplectisation of a contact 3-dimensional manifold. Namely, the author studies the boundary value problem on pseudoholomorphic curves of finite energy with appropriately prescribed ends, which belong to some totally real submanifolds.
The above problem arises in counting periodic orbits of the Reeb vector field and characteristic chords (i.e., segments of trajectories of the Reeb vector field connecting different points on a pre-fixed Legendrian submanifold), which is basic for relative contact homology (from the symplectic field theory due to Eliashberg, Givental and Hofer).
This paper handles local existence and embedding properties of pseudoholomorphic strips with mixed boundary conditions. Further, the author proves the compactness property and intersection result.
Part I, cf. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 21, No. 2, 139–185 (2004; Zbl 1073.53105)]; Part II, Commun. Pure Appl. Math. 57, No. 1, 1–58 (2004; Zbl 1073.53104).

MSC:

53D10 Contact manifolds (general theory)
35J60 Nonlinear elliptic equations
53D35 Global theory of symplectic and contact manifolds
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
35C05 Solutions to PDEs in closed form
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