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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Hulls of deformations in $ {\bf C}\sp{n}$


Author: H. Alexander
Journal: Trans. Amer. Math. Soc. 266 (1981), 243-257
MSC: Primary 32E20
DOI: https://doi.org/10.1090/S0002-9947-1981-0613794-X
MathSciNet review: 613794
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Abstract: A problem of $ {\text{E}}$. Bishop on the polynomially convex hulls of deformations of the torus is considered. Let the torus $ {T^2}$ be the distinguished boundary of the unit polydisc in $ {{\mathbf{C}}^2}$. If $ t \mapsto T_t^2$ is a smooth deformation of $ {T^2}$ in $ {{\mathbf{C}}^2}$ and $ {g_0}$ is an analytic disc in $ {{\mathbf{C}}^2}$ with boundary in $ {T^2}$, a smooth family of analytic discs $ t \mapsto {g_t}$, is constructed with the property that the boundary of $ {g_t}$ lies in $ T_t^2$. This construction has implications for the polynomially convex hulls of the tori $ T_t^2$. An analogous problem for a $ 2$-sphere in $ {{\mathbf{C}}^2}$ is also considered.


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DOI: https://doi.org/10.1090/S0002-9947-1981-0613794-X
Article copyright: © Copyright 1981 American Mathematical Society