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Polynomial hulls of sets fibered over the circle. (English) Zbl 0647.32017
Let \(D\subset {\mathbb{C}}\) be the unit disc, \(T=\partial D\) the unit circle, and \(\pi\) : \({\mathbb{C}}\) \(2\to {\mathbb{C}}\) the projection \(\pi (\zeta,z)=\zeta\). Let \(M\subset T\times {\mathbb{C}}\) be a compact, two- dimensional real submanifold of class C k (k\(\geq 2)\) such that the fiber \(M_{\zeta}=\{z\in {\mathbb{C}}:\) (\(\zeta\),z)\(\in M\}\) bounds a simply connected region \(Y_{\zeta}\subset {\mathbb{C}}\) containing the origin for each \(\zeta\) \(\in T\). Assume in addition that M is totally real in \({\mathbb{C}}^ 2;\) equivalently, \(\pi\) : \(M\to T\) is a submersion.
It is proved that for each point (a,b) in the polynomial hull \^M of M (a\(\in D)\) there is a function f, holomorphic in D and smooth of order \(C^{k-0}\) on \=D, such that \(f(a)=b\) and \(f(\zeta)\in M_{\zeta}\) for each \(\zeta\) \(\in T\). If (a,b) is in the boundary \(\Sigma =\partial \hat M\cap \pi ^{-1}(D)\), then such an f is unique, and \(\Sigma\) is a Levi- flat hypersurface of class \(C^{k-1}\). The pair (\(\Sigma\),M) is a hypersurface with boundary of class \(C^{k-2}.\)
This gives a precise description of the polynomial hull of M, and it also solves the Hilbert boundary problem with the data \(\{M_{\zeta}:\) \(\zeta\) \(\in T\}\). For related results see the papers of H. Alexander and J. Wermer [Math. Ann. 271, 99-109 (1985; Zbl 0538.32011)], Z. Slodkowski [Proc. Am. Math. Soc. 96, 255-260 (1986; Zbl 0588.32017)], and the new paper by Z. Slodkowsi [Polynomial hulls in \({\mathbb{C}}^ 2 \)and quasicircles, to appear].
Reviewer: F.Forstnerič

32E20 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
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