zbMATH — the first resource for mathematics

On the structure of pseudo-holomorphic discs with totally real boundary conditions. (English) Zbl 0931.53023
We study the structure of \(J\)-holomorphic discs in relation to the Fredholm theory of pseudo-holomorphic discs with totally real boundary conditions in almost complex manifolds \((M,J)\). We prove that any \(J\)-holomorphic disc with totally real boundary condition that is injective in the interior except at a discrete set of points, which we call a “normalized disc”, must either have some boundary point that is regular and has multiplicity one, or satisfy that its image forms a smooth immersed compact surface (without boundary) with a finite number of self-intersections and a finite number of branch points.
In the course of proving this theorem, we also prove several theorems on the local structure of boundary points of \(J\)-holomorphic discs, and, as an application, we give the first complete treatment of the transversality result for Floer’s pseudo-holomorphic trajectories for Lagrangian intersections in symplectic geometry.
Reviewer: Y.-G.Oh (Madison)

53C40 Global submanifolds
32Q65 Pseudoholomorphic curves
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Full Text: DOI
[1] Aronszajin, N. A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order.J. Math. Pure Appl. 36(9), 232–249 (1957).
[2] Chirka, E. M. Regularity of the boundaries of analytic sets.Math. USSR Sbornik 45 291–335 (1983). · Zbl 0525.32005 · doi:10.1070/SM1983v045n03ABEH001010
[3] Floer, A. The unregularized gradient flow of the symplectic action.Comm. Pure Appl. Math. 41, 774–813 (1988). · Zbl 0633.53058
[4] Floer, A., Hofer, H., and Salamon, D. Transversality in elliptic Morse theory for the symplectic action.Duke Math. J. 80, 251–292 (1995). · Zbl 0846.58025 · doi:10.1215/S0012-7094-95-08010-7
[5] Gromov, M. Pseudo-holomorphic curves in symplectic manifolds.Invent. Math. 81, 307–347 (1985). · Zbl 0592.53025 · doi:10.1007/BF01388806
[6] Globevnik, J., and Stout, E. Analytic discs with rectifiable simple closed curves as ends.Ann. Math. 127, 380–401 (1988). · Zbl 0644.32003 · doi:10.2307/2007060
[7] Hofer, H., and Wysocki, K. First order elliptic systems and the existence of homoclinic orbits in Hamiltonian systems.Math. Ann. 288, 483–503 (1990). · Zbl 0702.34039 · doi:10.1007/BF01444543
[8] Kwon, D., and Oh, Y.-G. Structure of the image of (pseudo)-holomorphic discs with totally real boundary condition. Preprint, 1996. · Zbl 0951.32025
[9] Lalonde, F. Hamiltonian collapsing of irrational Lagrangian submanifolds with small first Betti-number.Commun. Math. Phys. 149, 613–622 (1992). · Zbl 0806.58020 · doi:10.1007/BF02096945
[10] McDuff, D. Examples of symplectic structures.Invent. Math. 89, 13–36 (1987). · Zbl 0625.53040 · doi:10.1007/BF01404672
[11] McDuff, D. The local behavior of holomorphic curves in almost complex 4-manifolds.J. Diff. Geom. 34, 143–164 (1991). · Zbl 0736.53038
[12] McDuff, D. Singularities of J-holomorphic curves in almost complex 4-manifolds.J Geom. Anal. 2, 249–266 (1992). · Zbl 0758.53019 · doi:10.1007/BF02921295
[13] Oh, Y.-G. Removal of boundary singularities of pseudo-holomorphic curves with Lagrangian boundary conditions.Comm. Pure Appl. Math. 45, 121–139 (1992). · Zbl 0743.58018 · doi:10.1002/cpa.3160450106
[14] Oh, Y.-G. Floer cohomology of Lagrangian intersections and pseudo-holomorphic discs, I.Comm. Pure Appl. Math. 46, 949–994 (1993). · Zbl 0795.58019 · doi:10.1002/cpa.3160460702
[15] Oh, Y.-G. Fredholm theory of holomorphic discs under the perturbation of boundary conditions.Math. Z. 222 (1996), 505–520. · Zbl 0863.53024 · doi:10.1007/BF02621880
[16] Polterovich, L. The Maslov class of Lagrange surface and Gromov’s pseudo-holomorphic curves.Trans. A.M.S. 325, 241–248 (1991). · Zbl 0719.53016
[17] Parker, H., and Wolfson, J. Pseudo-holomorphic maps and bubble trees.J. Geom. Anal. 3, 63–97 (1993). · Zbl 0759.53023 · doi:10.1007/BF02921330
[18] Sikorav, J. C. Quelques propriétés des plongements lagrangiens. Preprint, Université Paris-Sud. Orsay, 1990.
[19] Stout, E., Bounded extensions: The case of discs in polydiscs.J. Analyse Math. 10, 239–254 (1975). · Zbl 0317.32015 · doi:10.1007/BF02786814
[20] Ye, R. Gromov’s compactness theorem for pseudo-holomorphic curves.Trans. Amer. Math. Soc. 342(2), 671–694 (1994). · Zbl 0810.53024
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.