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Perturbing analytic discs attached to maximal real submanifolds of \(\mathbb{C}^ N\). (English) Zbl 0861.32013
Let \(f\) be an analytic disc in \(\mathbb{C}^N\) attached to a maximal real submanifold \(M\) of \(\mathbb{C}^N\). The author introduced in a recent paper [Math. Z. 217, No. 2, 287-316 (1994; Zbl 0806.58044)] partial indices \(k_j\), \(1\leq j\leq N\), of \(M\) along the boundary of \(f\) and showed that if \(k_j\geq 0\) for all \(j\) then the family of nearby analytic discs attached to \(M\) depends on \(k_1+ \cdots +k_N\) parameters. Y.-G. Oh sharpened this by proving the same when \(k_j\geq -1\) for all \(j\) and showed that in terms of stability this is the best possible condition. In the paper under review the author explains why the latter condition is natural and give a simple proof of Oh’s result in the orientable case.

32G10 Deformations of submanifolds and subspaces
32V40 Real submanifolds in complex manifolds
32D10 Envelopes of holomorphy
Full Text: DOI
[1] Baouendi, M.S., L.P. Rothschild and J.-M. Trepreau — On the geometry of analytic discs attached to real manifolds. Preprint. · Zbl 0821.32014
[2] Birkhoff, G.D, On a simple type of irregular singular point, Trans. amer. math. soc., 14, 463, (1913)
[3] Cartan, H, Calcul differentiel, (1967), Hermann Paris · Zbl 0156.36102
[4] Chirka, E.M; Chirka, E.M, Regularity of boundaries of analytic sets, Mat. sb. (N.S.), Math. USSR sb., 45, 159, 291-335, (1983), English translation in · Zbl 0525.32005
[5] C̆erne, M. — Thesis, University of Wisconsin-Madison. In preparation.
[6] Forstneric̆, F, Analytic discs with boundaries in a maximal real submanifold of C2, Ann. inst. Fourier, 37, 1-44, (1987) · Zbl 0583.32038
[7] Globevnik, J, Perturbation by analytic discs along maximal real submanifolds of CN, Math. Z., 217, 287-316, (1994) · Zbl 0806.58044
[8] Hill, D.C; Taiani, G, Families of analytic discs in Cn with boundaries in a prescribed CR manifold, Ann. scuola norm. sup. Pisa, 5, 327-380, (1978) · Zbl 0399.32008
[9] Lang, S, Introduction to differentiable manifolds, (1962), Interscience New York, London · Zbl 0103.15101
[10] Oh, Y.-G. — The Fredholm-regularity and realization of Riemann-Hilbert problem and application to the perturbation theorem of analytic discs. Preprint.
[11] Oh, Y.-G. — Fredholm theory of holomorphic discs with Lagrangian or totally real boundary conditions under the perturbation of boundary conditions. Preprint.
[12] Plemelj, J, Riemannsche funktionenscharen mit gegebener monodromiegruppe, Monatsh. math. phys., 19, 211-246, (1908) · JFM 39.0461.01
[13] Pressley, A; Segal, G, Loop groups, (1986), Oxford Science Publ., Clarendon Press Oxford · Zbl 0618.22011
[14] Vekua, N.P, Systems of singular integral equations, (1967), Nordhoff Groningen · Zbl 0166.09802
[15] Vekua, N.P, Systems of singular integral equations, (1970), Nauka Moscow, Russian
[16] Trepreau, J.-M. — On the global Bishop equation. In preparation.
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