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Fredholm theory of holomorphic discs under the perturbation of boundary conditions. (English) Zbl 0863.53024
We prove that on any (almost) Kähler manifold \((M,\omega,J)\) and for any given compact Lagrangian submanifold \(L\subset(M,\omega)\), there exists a dense subset \(({\mathcal D}^L_\omega(M))_{\text{reg}}\) of \({\mathcal D}_\omega(M)\) such that if \(\phi\in({\mathcal D}^L_\omega(M))_{\text{reg}}\), any (not multiply-covered) holomorphic disc with boundary on \(\phi(L)\) in Fredholm-regular. Here, \({\mathcal D}_\omega(M)\) is the set of Hamiltonian (or exact) diffeomorphisms on \((M,\omega)\). Using this and Gromov’s existence scheme, refined by Polterovich, of pseudo-holomorphic discs on \(\mathbb{C}^n\), we prove that for any given compact Lagrangian submanifold \(L\subset\mathbb{C}^n\) and for \(\phi\in({\mathcal D}^L_\omega(\mathbb{C}^n))_{\text{reg}}\), there exists a Fredholm-regular holomorphic (not just \(J\)-holomorphic for some \(J\)!) disc with respect to the standard complex structure on \(\mathbb{C}^n\), \(w:(D^2,\partial D^2)\to(\mathbb{C}^n,\phi(L))\) such that its Maslov index \(\mu_{\phi(L)}(w)\) satisfies \(3-n\leq \mu_{\phi(L)}(w)\leq n+1\). This is a purely complex analytic analogue of a result by Polterovich. We prove similar results under the perturbation by Lagrangian isotopies. Finally, we also develop a Fredholm theory and prove a similar perturbation result for holomorphic discs with totally real boundary conditions in an (almost) complex manifold \((M,J)\).

53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
32C25 Analytic subsets and submanifolds
53C55 Global differential geometry of Hermitian and Kählerian manifolds
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53C40 Global submanifolds
32Q99 Complex manifolds
47A53 (Semi-) Fredholm operators; index theories
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[1] Arnold, V.I.: On a characteristic class entering the quantization conditions. Funct. Anal. Appl.1, 1–14 (1967) · Zbl 0175.20303 · doi:10.1007/BF01075861
[2] Aronszajin, N.: A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order. J. Math. Pure Appl. (9),36, 232–249 (1957)
[3] Floer, A.: The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math.41, 775–813 (1988) · Zbl 0633.53058 · doi:10.1002/cpa.3160410603
[4] Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds.81, 307–347 (1985) · Zbl 0592.53025
[5] Kobayashi, S, Nomizu, K.: Foundations of Differential Geometry. v. II, Wiley, New York, 1969 · Zbl 0175.48504
[6] 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
[7] Lions, J.L., Magenes, E.: Non-homogeneous Boundary Value Problems and Applications. Vol. 1, Springer-Verlag, New York/Heidelberg/Berlin, 1972 · Zbl 0227.35001
[8] Lê, H.V., Ono, K.: Symplectic fixed points, the Calabi invariant and Novikov homology. Max-Planck-Institut, Bonn. Preprint · Zbl 0822.58019
[9] McDuff, D.: Examples of symplectic structures. Invent. Math.89, 13–36 (1987) · Zbl 0625.53040 · doi:10.1007/BF01404672
[10] 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
[11] Oh, Y.-G.: On the structure of pseudo-holomorphic discs with totally real boundary conditions. To appear in J. Geom. Anal.
[12] Oh, Y.-G.: Riemann-Hilbert problem and application to the perturbation theory of analytic discs. Kyungpook Math. J.35, 39–75 (1995) · Zbl 0853.32017
[13] Polterovich, L.: The Maslov class of Lagrange Surface and Gromov’s pseudoholomorphic curves. Trans. A.M.S.325, 241–248 (1991) · Zbl 0719.53016
[14] Polterovich, L.: Symplectic displacement energy for Lagrangian submanifolds, Ergod. Th. and Dynam. Sys.13, 357–367 (1993) · Zbl 0781.58007 · doi:10.1017/S0143385700007410
[15] Sikorav, J.C.: Quelques propriétés des plongements lagrangiens. Université Paris-Sud, Orsay, 1990, Preprint
[16] Viterbo, C.: Intersection de sur-variétés Lagrangiennes, Fonctíonelles d’action et indices des systems Hamiltoniene, Bull. Soc. Math. Frances115, 361–380 (1987)
[17] Weinstein, A.: Lagrangian submanifolds and hamiltonian systems. Ann. Math.98, 377–410 (1973) · Zbl 0271.58008 · doi:10.2307/1970911
[18] Weinstein, A.: Connections of Berry and Hannay type for moving Lagrangian submanifolds. Advances in Math.82, 133–159 (1990) · Zbl 0713.58015 · doi:10.1016/0001-8708(90)90086-3
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