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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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On smooth and analytic disks in $\mathbf {C}^2$ with common boundary
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by Kang-Tae Kim PDF
Proc. Amer. Math. Soc. 122 (1994), 541-544 Request permission

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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Additional Information
  • © Copyright 1994 American Mathematical Society
  • 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