Polynomial hull of a sphere imbedded in C2

1991 ◽  
Vol 49 (1) ◽  
pp. 89-93 ◽  
Author(s):  
N. V. Shcherbina
Keyword(s):  
Polynomial Hull

Quasiconformal Contactomorphisms and Polynomial Hulls with Convex Fibers

1999 ◽  
Vol 51 (5) ◽  
pp. 915-935 ◽  
Author(s):  
Zoltán M. Balogh ◽  
Christoph Leuenberger
Keyword(s):  
Unit Circle ◽  
Complex Flow ◽  
One Dimensional ◽  
Holomorphic Motions ◽  
Strictly Convex ◽  
Fixed Domain ◽  
Polynomial Hulls ◽  
Polynomial Hull ◽  
The One ◽  
Special Case

AbstractConsider the polynomial hull of a smoothly varying family of strictly convex smooth domains fibered over the unit circle. It is well-known that the boundary of the hull is foliated by graphs of analytic discs. We prove that this foliation is smooth, and we show that it induces a complex flow of contactomorphisms. These mappings are quasiconformal in the sense of Korányi and Reimann. A similar bound on their quasiconformal distortion holds as in the one-dimensional case of holomorphic motions. The special case when the fibers are rotations of a fixed domain in C2 is studied in details.

scholarly journals DE PAEPE'S DISC HAS NONTRIVIAL POLYNOMIAL HULL

2002 ◽  
Vol 34 (04) ◽  
pp. 490-494 ◽  
Author(s):  
A. G. O'FARRELL ◽  
M. A. SANABRIA-GARCÍA
Keyword(s):  
Polynomial Hull

scholarly journals The polynomial hull of a thin two-manifold

1971 ◽  
Vol 38 (2) ◽  
pp. 377-389 ◽  
Author(s):  
Michael Freeman
Keyword(s):  
Polynomial Hull

scholarly journals Pointwise bounded approximation by polynomials

1992 ◽  
Vol 112 (1) ◽  
pp. 147-155 ◽  
Author(s):  
Anthony G. O'Farrell ◽  
Fernando Perez-Gonzalez
Keyword(s):  
Analytic Functions ◽  
Unit Disc ◽  
Open Unit ◽  
Open Unit Disc ◽  
Bounded Analytic Functions ◽  
Unbounded Component ◽  
Open Set ◽  
Approximation By Polynomials ◽  
Polynomial Hull ◽  
Image Position

For a bounded open set U ⊂ ℂ, we denote by H∞(U) the collection of all bounded analytic functions on U. We let X denote bdy (U), the boundary of U, Y denote the polynomial hull of U (the complement of the unbounded component of ℂ / X), and U* denote mt (Y), the interior of Y. We denote the sup norm of a function f: A → ℂ by ∥f∥A:We denote the space of all analytic polynomials by ℂ[z], and we denote the open unit disc by D and the unit circle by S1.

STATIONARY DISCS OF FIBRATIONS OVER THE CIRCLE

1995 ◽  
Vol 06 (06) ◽  
pp. 805-823 ◽  
Author(s):  
MIRAN ČERNE
Keyword(s):  
Unit Circle ◽  
Pseudoconvex Domains ◽  
Kobayashi Metric ◽  
Strictly Convex ◽  
Strongly Pseudoconvex Domains ◽  
Partial Indices ◽  
Stationary Disc ◽  
Polynomial Hull

Stationary discs of fibrations over the unit circle ∂D are considered. It is shown that if all fibers of a fibration Σ⊆∂D×Cn over the unit circle ∂D are strongly pseudoconvex hypersurfaces in Cn, then for every stationary disc f of the fibration Σ one can define the partial indices of f. In the case all fibers of Σ are strictly convex, it is proved that all partial indices of a stationary disc f are 0. It is also proved that in the case a stationary disc f of the fibration Σ is non-degenerate, the only possible partial indices of f are 0, 1 and –1. In particular, these results give information on the polynomial hull of Σ and new proofs of results related to the smoothness of the Kobayashi metric on some strongly pseudoconvex domains in Cn.

scholarly journals Smooth families of fibrations and analytic selections of polynomial hulls

1995 ◽  
Vol 52 (1) ◽  
pp. 97-105 ◽  
Author(s):  
Miran Černe
Keyword(s):  
Unit Circle ◽  
Complex Geometry ◽  
Continuity Method ◽  
Polynomial Hulls ◽  
Polynomial Hull ◽  
Selection Of

Constructed are strictly increasing smooth families Σt ⊆ ∂D × C2, t ∈ [0, 1], of fibrations over the unit circle with strongly pseudoconvex fibers all diffeomorphic to the ball such that there is no analytic selection of the polynomial hull of Σ0 and which end at the product fibration . In particular these examples show that the continuity method for describing the polynomial hull of a fibration over ∂D fails even if the complex geometry of the fibers is relatively simple.

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