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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 \((M,J)\).

MSC:
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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