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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

##### MSC:
 3.2e+21 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
##### Keywords:
analytic disc; polynomial hull; totally real submanifold
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