# zbMATH — the first resource for mathematics

Polynomial hulls in $${\mathbb{C}}^ 2$$ and quasicircles. (English) Zbl 0703.32007
For compact $$X\subset {\mathbb{C}}^ n$$ denote by $$\hat X$$ its polynomial hull. The main result of the paper is:
Theorem 1.1. Let $$X\subset \partial D\times {\mathbb{C}}$$ be a compact set such that for every $$\zeta\in \partial D$$, $$X(\zeta)=\{w\in {\mathbb{C}}:$$ ($$\zeta$$,w)$$\in X\}$$ is a simply connected continuum; here D is a unit disc in $${\mathbb{C}}$$. Assume that $$\hat X\setminus X$$ is nonempty. Then $$\hat X\setminus X$$ is equal to the union of the graphs of all $$H^{\infty}(D)$$ functions whose cluster values at $$\zeta\in \partial D$$ belong to X($$\zeta$$). (Below, an analytic disc will always denote such a graph). Furthermore,
(a) the relative boundary of $$\hat X\cap D\times {\mathbb{C}}$$, denoted by S, is covered by analytic disks, and every two distinct analytic discs contained in S are disjoint;
(b) every analytic disc contained in $$\hat X$$ and interesting S must be fully contained in S;
(c) for every $$z\in D$$ the fiber $$Y(z)=\{w\in {\mathbb{C}}:$$ (z,w)$$\in \hat X\}$$, of $$\hat X,$$ is a simply connected continuum and its topological boundary (in $${\mathbb{C}})$$ is equal to the fiber of S, i.e. $$S(z)=\{w\in {\mathbb{C}}:$$ (z,w)$$\in S\}$$.
Reviewer: S.M.Ivashkovich

##### MSC:
 3.2e+21 Polynomial convexity, rational convexity, meromorphic convexity in several complex variables
##### Keywords:
polynomial hull; analytic disc
Full Text:
##### References:
 [1] H. Alexander , Hulls of deformations in Cn , Trans. Amer. Math. Soc. 266 ( 1981 ), 243 - 257 . MR 613794 | Zbl 0493.32017 · Zbl 0493.32017 · doi:10.2307/1998396 [2] H. Alexander - J. Wermer , Polynomial hulls with convex fibers , Math. Ann. 27 ( 1985 ), 99 - 109 . MR 779607 | Zbl 0538.32011 · Zbl 0538.32011 · doi:10.1007/BF01455798 · eudml:163955 [3] E. Bedford - B. Gaveau , Envelopes of holomorphy of certain 2-spheres in C 2, Amer. J. Math . 105 ( 1983 ), 975 - 1009 . MR 708370 | Zbl 0535.32008 · Zbl 0535.32008 · doi:10.2307/2374301 [4] J. Bence - J.W. Helton - D.E. Marshall , Optimization over H\infty . [5] E. Bishop , Differentiable manifolds in complex Euclidean spaces , Duke Math. J. 32 ( 1965 ), 1 - 21 . Article | MR 200476 | Zbl 0154.08501 · Zbl 0154.08501 · doi:10.1215/S0012-7094-65-03201-1 · minidml.mathdoc.fr [6] E.M. Čirka , Regularity of boundaries of analytic sets , Math. USSR-Sb. 117 ( 1982 ), 291 - 334 , (Russian) Math. USSR-Sb. 45 ( 1983 ), 291 - 336 . MR 648411 | Zbl 0525.32005 · Zbl 0525.32005 · doi:10.1070/SM1983v045n03ABEH001010 [7] F Forstnerič , Polynomially convex hulls with piecewise smooth boundaries , Math. Ann. 276 ( 1986 ), 97 - 104 . MR 863710 | Zbl 0585.32016 · Zbl 0585.32016 · doi:10.1007/BF01450928 · eudml:164187 [8] F Forstnerič , Polynomial hulls of sets fibered over the circle , Indiana Univ. Math. J. 37 ( 1988 ), 869 - 889 . MR 982834 | Zbl 0647.32017 · Zbl 0647.32017 · doi:10.1512/iumj.1988.37.37042 [9] J.W. Helton - R.E. Howe , A bang-bang theorem for optimization over spaces of analytic functions , J. Approx. Theory 47 ( 1986 ), 101 - 121 . MR 844946 | Zbl 0589.49004 · Zbl 0589.49004 · doi:10.1016/0021-9045(86)90036-5 [10] J.W. Helton - D.F. Schwartz - S.E. Warchawski , Local optima in H\infty produce a constant objective function , Complex Variables , 8 ( 1987 ), 65 - 81 . Zbl 0577.49009 · Zbl 0577.49009 [11] O. Lehto , University functions and Teichmüller spaces , Springer-Verlag , New York 1986 . MR 486503 [12] Chr. Pommerenke , Univalent functions, Vandenhoeck and Ruprecht in Göttingen , 1975 . MR 507768 | Zbl 0298.30014 · Zbl 0298.30014 [13] Z. Slodkowski , Local maximum property and q-plurisubharmonic functions in uniform algebras , J. Math Anal. Appl. 115 ( 1986 ). 105 - 130 . MR 835588 | Zbl 0646.46047 · Zbl 0646.46047 · doi:10.1016/0022-247X(86)90027-2 [14] Z. Slodkowski , Polynomially convex hulls with convex sections and interpolating spaces , Proc. Amer. Math. Soc. 96 ( 1986 ), 255 - 260 . MR 818455 | Zbl 0588.32017 · Zbl 0588.32017 · doi:10.2307/2046164 [15] J. Wermer , Polynomially convex hulls and analyticity , Ark. Mat. , 20 ( 1982 ), 129 - 135 . MR 660131 | Zbl 0491.32013 · Zbl 0491.32013 · doi:10.1007/BF02390504
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.