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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



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
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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Article copyright: © Copyright 1994 American Mathematical Society