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Analytic disks with boundaries in a maximal real submanifold of $${\mathbb C}^ 2$$. (English) Zbl 0583.32038
Let M be an orientable, smoothly embedded, totally real submanifold of $${\mathbb{C}}^ 2$$. A continuous map F: $$\bar D\to {\mathbb{C}}^ 2$$ of the closed unit disk $$\bar D=\{| z| \leq 1\}$$ into $${\mathbb{C}}^ 2$$ that is holomorphic on the open disk $$D=\{| z| <1\}$$ and maps its boundary $$bD=\{| z| =1\}$$ into M is called an analytic disk with boundary in M. Given such a disk $$F^ 0$$ we give a rather complete description of the nearby disks with boundary in M in terms of an integer $$m=Ind_{F^ 0}M$$, called the index of M along $$F^ 0$$, which only depends on the homology class of the path $$F^ 0: bD\to M$$ and is easily calculated from the local parametrization of M along $$F^ 0(bD)$$. If $$m\geq 1$$, there is a $$2m+2$$ parameter family of nearby disks which are stable under small deformations of M in $${\mathbb{C}}^ 2$$. If $$m\leq 0$$, there are no nearby disks, and the initial disk may disappear under a small deformation of M. When $$m=1$$ the nearby disks form a smooth hypersurface with boundary in M [this is related to a result of E. Bedford in Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 9, 529-570 (1982; Zbl 0574.32019)]. We also prove a regularity theorem for immersed families of analytic disks with boundary in M, consider several examples, and construct a totally real three-torus in $${\mathbb{C}}^ 3$$ with a bizzare polynomially convex hull.

MSC:
 32B15 Analytic subsets of affine space 32V40 Real submanifolds in complex manifolds 32G10 Deformations of submanifolds and subspaces 32A40 Boundary behavior of holomorphic functions of several complex variables 32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
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References:
 [1] H. ALEXANDER, Hulls of deformations in cn, Trans. Amer. Math. Soc., 266 (1981), 243-257. · Zbl 0493.32017 [2] H. ALEXANDER, A note on polynomially convex hulls, Proc. Amer. Math. Soc., 33 (1972), 389-391. · Zbl 0239.32013 [3] H. ALEXANDER and J. WERMER, Polynomial hulls with convex fibers, Math. Ann., 271 (1985), 99-109. · Zbl 0538.32011 [4] E. BEDFORD, Stability of the polynomial hull of T2, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 8 (1982), 311-315. · Zbl 0472.32012 [5] E. BEDFORD, Levi flat hypersurfaces in C2 with prescribed boundary : stability, Annali Scuola Norm. Sup. Pisa cl. Sci., 9 (1982), 529-570. · Zbl 0574.32019 [6] E. BEDFORD and B. GAVEAU, Envelopes of holomorphy of certain two-spheres in C2, Amer. J. Math., 105 (1983), 975-1009. · Zbl 0596.32019 [7] E. BISHOP, Differentiable manifolds in complex Euclidean spaces, Duke Math. J., 32 (1965), 1-21. · Zbl 0154.08501 [8] A. BOGGES and J. PITTS, CR extensions near a point of higher type, Duke Math. J., 52 (1985), 67-102. · Zbl 0573.32019 [9] A. BROWDER, Cohomology of maximal ideal spaces, Bull Amer. Math. Soc., 67 (1961), 515-516. · Zbl 0107.09501 [10] H. CARTAN, Calcul différentiel, Hermann, Paris 1967. · Zbl 0156.36102 [11] S. CHERN and E. SPANIER, A theorem on orientable surfaces in four-dimensional space, Comm. Math. Helv., 25 (1951), 205-209, North Holland, Amsterdam 1975. · Zbl 0043.38403 [12] E.M. CIRKA, Regularity of boundaries of analytic sets, (Russian) Math. Sb (N.S.) 117 (159), (1982), 291-334. English translation in Math. USSR Sb., 45 (1983), 291-336. · Zbl 0525.32005 [13] F. DOCQUIER and H. GRAUERT, Levisches problem und rungescher satz für teilgebiete steinscher mannigfaltigkeiten, Math. Ann., 140 (1960), 94-123. · Zbl 0095.28004 [14] T. DUCHAMP and E.L. STOUT, Maximum modulus sets, Ann. Inst. Fourier, 31-3 (1981), 37-69. · Zbl 0439.32007 [15] P.L. DUREN, The theory of hp spaces, Academic Press, New-York and London, 1970. · Zbl 0215.20203 [16] M. GOLUBITSKY and V. GUILLEMIN, Stable mappings and their singularities, Graduate Texts in Mathematics, 41, Springer-Verlag, New-York, Heidelberg, Berlin 1973. · Zbl 0294.58004 [17] F.R. HARVEY and R.O. WELLS, Holomorphic approximation and hyperfunction theory on a C1 totally real submanifold of a complex manifold, Math. Ann., 197 (1972), 287-318. · Zbl 0246.32019 [18] D. HILBERT, Grundzüge einer allgemeiner theorie der linearen integralgleichungen, Leipzig, 1912. [19] D.C. HILL and G. TAIANI, Families of analytic disks in cn with boundaries in a prescribed CR manifold, Ann. Scuola Norm. Sup. Pisa, 5 (1978), 327-380. · Zbl 0399.32008 [20] C.E. KENIG and S.M. WEBSTER, The local hull of holomorphy of a surface in the space of two complex variables, Invent. Math., 67 (1982), 1-21. · Zbl 0489.32007 [21] C.E. KENIG and S.M. WEBSTER, On the hull of holomorphy of n-manifold in cn, Annali Scuola Norm. Sup. Pisa sci., 11 (1984), 261-280. · Zbl 0558.32006 [22] L. LEMPERT, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France, 109 (1981), 427-474. · Zbl 0492.32025 [23] W. POGORZELSKI, Integral equations and their applications, Pergamon Press, Oxford, 1966. · Zbl 0137.30502 [24] W. RUDIN, Totally real Klein bottles in C2, Proc. Amer. Math. Soc., 82 (1981), 653-654. · Zbl 0483.32013 [25] N. STEENROD, The topology of fiber bundles, Princeton University Press, Princeton, New Jersey, 1951. · Zbl 0054.07103 [26] G. STOLZENBERG, A hull with no analytic structure, J. Math. Mech., 12 (1963), 103-112. · Zbl 0113.29101 [27] S. WEBSTER, Minimal surfaces in Kähler manifolds, Preprint. · Zbl 0561.53054 [28] S. WEBSTER, The Euler and pontrjagin numbers of an n-manifold in cn, Preprint. · Zbl 0566.32015 [29] A. WEINSTEIN, Lectures on symplectic manifolds, Regional Conference Series in Mathematics 29, Amer. Math. Soc., Providence, R.I., 1977. · Zbl 0406.53031 [30] J. WERMER, Polynomially convex hulls and analyticity, J. Math. Mech., 20 (1982), 129-135. · Zbl 0491.32013 [31] L.V. WOLFERSDORF, A class of nonlinear Riemann-Hilbert problems for holomorphic functions, Math. Nachr., 116 (1984), 89-107. · Zbl 0554.30019 [32] F. FORSTNERIC, Polynomially convex hulls with piecewise smooth boundaries, Math. Ann., 276 (1986), 97-104. · Zbl 0585.32016 [33] F. FORSTNERIC, On the nonlinear Riemann - Hilbert problem. To appear. · Zbl 0739.32022
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