Analytic disks with boundaries in a maximal real submanifold of 𝐂 2
Annales de l'Institut Fourier, Volume 37 (1987) no. 1, pp. 1-44.

Let M be a two dimensional totally real submanifold of class C 2 in C 2 . A continuous map F:Δ ¯C 2 of the closed unit disk Δ ¯C into C 2 that is holomorphic on the open disk Δ and maps its boundary bΔ into M is called an analytic disk with boundary in M. Given an initial immersed analytic disk F 0 with boundary in M, we describe the existence and behavior of analytic disks near F 0 with boundaries in small perturbations of M in terms of the homology class of the closed curve F 0 (bΔ) in M. We also prove a regularity theorem for immersed families of analytic disks, consider several examples, and construct a totally real three torus in C 3 with a bizzare polynomially convex hull.

Soit M une sous-variété totalement réelle de dimension 2 et classe C 2 dans C 2 . Une application continue F:Δ ¯C 2 de disque-unité fermé Δ ¯C dans C 2 , qui est holomorphe sur Δ et applique sa frontière bΔ dans M, est appelée un disque analytique avec frontière dans M. Etant donné un disque initial F 0 avec frontière dans M, on détermine l’existence des disques près de F 0 avec les frontières dans les petites perturbations de M à l’aide de la classe d’homologie de courbe F 0 (bΔ) dans M. On démontre aussi un théorème de régularité pour des familles des disques et on construit un tore totalement réel de dimension 3 dans C 3 avec une étrange enveloppe convexe polynomiale.

@article{AIF_1987__37_1_1_0,
     author = {Forstneric, Franc},
     title = {Analytic disks with boundaries in a maximal real submanifold of ${\bf C}^2$},
     journal = {Annales de l'Institut Fourier},
     pages = {1--44},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {37},
     number = {1},
     year = {1987},
     doi = {10.5802/aif.1076},
     zbl = {0583.32038},
     mrnumber = {88j:32019},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.1076/}
}
TY  - JOUR
AU  - Forstneric, Franc
TI  - Analytic disks with boundaries in a maximal real submanifold of ${\bf C}^2$
JO  - Annales de l'Institut Fourier
PY  - 1987
SP  - 1
EP  - 44
VL  - 37
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.1076/
DO  - 10.5802/aif.1076
LA  - en
ID  - AIF_1987__37_1_1_0
ER  - 
%0 Journal Article
%A Forstneric, Franc
%T Analytic disks with boundaries in a maximal real submanifold of ${\bf C}^2$
%J Annales de l'Institut Fourier
%D 1987
%P 1-44
%V 37
%N 1
%I Institut Fourier
%C Grenoble
%U https://aif.centre-mersenne.org/articles/10.5802/aif.1076/
%R 10.5802/aif.1076
%G en
%F AIF_1987__37_1_1_0
Forstneric, Franc. Analytic disks with boundaries in a maximal real submanifold of ${\bf C}^2$. Annales de l'Institut Fourier, Volume 37 (1987) no. 1, pp. 1-44. doi : 10.5802/aif.1076. https://aif.centre-mersenne.org/articles/10.5802/aif.1076/

[1] H. Alexander, Hulls of deformations in Cn, Trans. Amer. Math. Soc., 266 (1981), 243-257. | MR | Zbl

[2] H. Alexander, A note on polynomially convex hulls, Proc. Amer. Math. Soc., 33 (1972), 389-391. | MR | Zbl

[3] H. Alexander and J. Wermer, Polynomial hulls with convex fibers, Math. Ann., 271 (1985), 99-109. | MR | Zbl

[4] E. Bedford, Stability of the polynomial hull of T2, Ann. Scuola Norm. Sup. Pisa Cl. Sci., (4) 8 (1982), 311-315. | Numdam | Zbl

[5] E. Bedford, Levi flat hypersurfaces in C2 with prescribed boundary : Stability, Annali Scuola Norm. Sup. Pisa cl. Sci., 9 (1982), 529-570. | Numdam | MR | Zbl

[6] E. Bedford and B. Gaveau, Envelopes of holomorphy of certain two-spheres in C2, Amer. J. Math., 105 (1983), 975-1009. | MR | Zbl

[7] E. Bishop, Differentiable manifolds in complex Euclidean spaces, Duke Math. J., 32 (1965), 1-21. | MR | Zbl

[8] A. Bogges and J. Pitts, CR extensions near a point of higher type, Duke Math. J., 52 (1985), 67-102. | Zbl

[9] A. Browder, Cohomology of maximal ideal spaces, Bull Amer. Math. Soc., 67 (1961), 515-516. | MR | Zbl

[10] H. Cartan, Calcul Différentiel, Hermann, Paris 1967. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[13] F. Docquier and H. Grauert, Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann., 140 (1960), 94-123. | MR | Zbl

[14] T. Duchamp and E.L. Stout, Maximum modulus sets, Ann. Inst. Fourier, 31-3 (1981), 37-69. | Numdam | MR | Zbl

[15] P.L. Duren, The Theory of Hp spaces, Academic Press, New-York and London, 1970. | MR | Zbl

[16] M. Golubitsky and V. Guillemin, Stable Mappings and their Singularities, Graduate Texts in Mathematics, 41, Springer-Verlag, New-York, Heidelberg, Berlin 1973. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[22] L. Lempert, La métrique de Kobayashi et la représentation des domaines sur la boule, Bull. Soc. Math. France, 109 (1981), 427-474. | Numdam | MR | Zbl

[23] W. Pogorzelski, Integral Equations and their Applications, Pergamon Press, Oxford, 1966. | MR | Zbl

[24] W. Rudin, Totally real Klein bottles in C2, Proc. Amer. Math. Soc., 82 (1981), 653-654. | MR | Zbl

[25] N. Steenrod, The Topology of Fiber Bundles, Princeton University Press, Princeton, New Jersey, 1951. | Zbl

[26] G. Stolzenberg, A hull with no analytic structure, J. Math. Mech., 12 (1963), 103-112. | MR | Zbl

[27] S. Webster, Minimal surfaces in Kähler manifolds, Preprint. | Zbl

[28] S. Webster, The Euler and Pontrjagin numbers of an n-manifold in Cn, Preprint. | Zbl

[29] A. Weinstein, Lectures on Symplectic Manifolds, Regional Conference Series in Mathematics 29, Amer. Math. Soc., Providence, R.I., 1977. | MR | Zbl

[30] J. Wermer, Polynomially convex hulls and analyticity, J. Math. Mech., 20 (1982), 129-135. | MR | Zbl

[31] L.V. Wolfersdorf, A class of nonlinear Riemann-Hilbert problems for holomorphic functions, Math. Nachr., 116 (1984), 89-107. | MR | Zbl

[32] F. Forstneric, Polynomially convex hulls with piecewise smooth boundaries, Math. Ann., 276 (1986), 97-104. | MR | Zbl

[33] F. Forstneric, On the nonlinear Riemann - Hilbert problem. To appear.

Cited by Sources: