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On smooth and analytic disks in $ \bold C\sp 2$ with common boundary


Author: Kang-Tae Kim
Journal: Proc. Amer. Math. Soc. 122 (1994), 541-544
MSC: Primary 32F99; Secondary 32D10
DOI: https://doi.org/10.1090/S0002-9939-1994-1204380-6
MathSciNet review: 1204380
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Abstract: We construct explicitly a real analytic embedded real two-dimensional disk in $ {{\mathbf{C}}^2}$ totally real except at exactly one elliptic complex tangent point, which shares the common boundary with an analytic disk in the same $ {{\mathbf{C}}^2}$, but does not contain this analytic disk in its envelope of holomorphy. The same proof further yields an explicit example of a holomorphic re-embedding of the standard two-sphere into $ {{\mathbf{C}}^2}$ in such a way that the new embedding shows some exceptional properties: It bounds a real three-dimensional Levi flat cell in $ {{\mathbf{C}}^2}$ foliated by analytic disks, which is not polynomially convex. In particular, this new embedding of the standard two-sphere cannot be a subset of any compact strongly pseudoconvex surface in $ {{\mathbf{C}}^2}$ or a subset of any strongly pseudoconvex graph in $ {{\mathbf{C}}^2}$ in the sense of Bedford and Gaveau.


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  • [1] E. Bedford, Levi flat hypersurfaces in $ {{\mathbf{C}}^2}$ with prescribed boundary: Stability, Ann. Scuola Norm. Sup. Pisa. Cl. Sci. (4) 9 (1982), 529-570. MR 693779 (85d:32029)
  • [2] E. Bedford and B. Gaveau, Envelope of holomorphy of certain 2-spheres in $ {{\mathbf{C}}^2}$, Amer. J. Math. 105 (1983), 957-1009. MR 708370 (84k:32016)
  • [3] E. Bedford and W. Klingenberg, On the envelope of holomorphy of a 2-sphere in $ {{\mathbf{C}}^2}$, Amer. Math. Soc. 4 (1991), 623-646. MR 1094437 (92j:32034)
  • [4] E. Bishop, Differentiate manifolds in complex Euclidean space, Duke Math. J. 32 (1965), 1-21. MR 0200476 (34:369)
  • [5] F. Forstnerič, Intersections of smooth and analytic disks, preprint.
  • [6] F. Forstnerič and T. Duchamp, Intersections of analytic and totally real discs, preprint.
  • [7] M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347. MR 809718 (87j:53053)
  • [8] R. Gunning and H. Rossi, Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, NJ, 1965. MR 0180696 (31:4927)
  • [9] M. Hirsch, Differential topology, Springer-Verlag, Heidelberg, Berlin, and New York, 1976. MR 0448362 (56:6669)
  • [10] L. R. Hunt, The local envelope of holomorphy of an n-manifold in $ {{\mathbf{C}}^n}$, Boll. Un. Mat. Ital. 4 (1971), 12-35. MR 0294690 (45:3758)
  • [11] C. Kenig and S. Webster, The local hull of holomorphy of a surface in the space of two complex variables, Invent. Math. 67 (1982), 1-21. MR 664323 (84c:32014)
  • [12] J. Wermer, On a domain equivalent to the bidisk, Math. Ann. 248 (1980), 193-194. MR 575937 (81g:32002)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1204380-6
Article copyright: © Copyright 1994 American Mathematical Society

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