On smooth and analytic disks in $\mathbf {C}^2$ with common boundary
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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.References
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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