A new class of polynomially convex sets

Forstnerič, Franc; Stout, Edgar Lee

doi:10.1007/bf02384330

Cited by 48 publications

(45 citation statements)

References 12 publications

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“…(1) tn the strictly pseudoconvex case. F. Forstneri~ and E. L. Stout [9] gave the remarkable generalization that the result remains true if we allow S to have finitely many isolated complex points of hyperbolic type (see Section 3 for the definition). A little later, J. Duval [6] published an elegant alternative proof, which works also for certain weakly convex domains.…”

Section: Theorem 1 Let D C C 2 Be a Relatively Corn Pact Domain Witmentioning

confidence: 95%

Totally real discs in non-pseudoconvex boundaries

Porten

2003

Ark. Mat.

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Abstract. Let D be a relatively compact domain in C 2 with smooth connected boundary OD. A compact set KCOD is called removable if any continuous CR function defined on OD\K admits a holomorphic extension to D. If D is strictly pseudoconvex, a theorem of B. J6ricke states that any compact K contained in a smooth totally real disc SCOD is removable. In the present article we ,show that this theorem is true without any assumption on pseudoconvexity.

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Section: Theorem 1 Let D C C 2 Be a Relatively Corn Pact Domain Witmentioning

confidence: 95%

Totally real discs in non-pseudoconvex boundaries

Porten

2003

Ark. Mat.

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“…We will abbreviate the phrase "point of complex tangency" to CR singularity. When S has an isolated CR singularity at p ∈ S and the order of contact of T p (S) with S at p equals 2, we now have a nearly complete understanding of the local polynomial hull of S at p. This knowledge stems from the works of Bishop [5], Forstnerič-Stout [10], and Jöricke [12]. Much less is known when the order of contact of T p (S) with S at an isolated CR singularity p is greater than 2.…”

Section: Then [Z F ] D = C(d) In Particular γ D (F ) Is Polynomiamentioning

confidence: 99%

Polynomial Approximation, Local Polynomial Convexity, and Degenerate Cr Singularities — Ii

Bharali

2011

Int. J. Math.

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Abstract. We provide some conditions for the graph of a Hölder-continuous function on D, where D is a closed disc in C, to be polynomially convex. Almost all sufficient conditions known to date -provided the function (say F ) is smooth -arise from versions of the Weierstrass Approximation Theorem on D. These conditions often fail to yield any conclusion if rank R DF is not maximal on a sufficiently large subset of D. We bypass this difficulty by introducing a technique that relies on the interplay of certain plurisubharmonic functions. This technique also allows us to make some observations on the polynomial hull of a graph in C 2 at an isolated complex tangency. Introduction and statement of results This paper has evolved from the following two considerations:• Let D be a closed disc in C and let F ∈ C(D). There are numerous results that provide sufficient conditions for the uniform algebra on D generated by z and F to equal C(D); see [21,19,16,9,17,3]. These conditions are sufficient, naturally, for the graph of F to be polynomially convex. The aforementioned results require -either by explicit fiat or through some a priori condition on F -that F −1 {F (ζ)} be at most countable for a.e. ζ ∈ D. This is troublesome because it excludes, for instance, C( ∼ = R 2 )-valued functions in C 1 (D) having rank R DF < 2 on a non-empty open subset of D. It would thus be useful to devise techniques that allow us to detect polynomial convexity without imposing such restrictions.• In a recent work, Dieu & Chi [8] employed a technique that can be applied to situations very different from the one that they study. Their idea, suitably adapted to the given context, might serve as quite a general tool in the study of polynomial convexity of graphs. We wish to argue this case by presenting a couple of adaptations of their idea. Given F as above and a set S ⊆ Dom(F ), we shall write Γ S (F ) := Graph(F ) ∩ (S × C). Let us understand some of the known sufficient conditions on F , for Γ D (F ) to be polynomially convex, by examining critically a representative result selected from the aforementioned papers. Hence consider: 2000 Mathematics Subject Classification. Primary 30E10, 32E20, 32F05.

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“…The point p is elliptic if λ < 1/2 and hyperbolic if λ > 1/2; the degenerate case λ = 1/2 does not arise for a generic M. At an elliptic point M has a nontrivial local envelope of holomorphy consisting of a family of small analytic discs (the so called Bishop discs, [3]). At a hyperbolic point M is locally holomorphically convex [24] and it admits a basis of tubular Stein neighborhoods [53].…”

Section: Remark 75mentioning

confidence: 99%

Stein structures and holomorphic mappings

Forstnerič

Slapar

2007

Math. Z.

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We prove that every continuous map from a Stein manifold X to a complex manifold Y can be made holomorphic by a homotopic deformation of both the map and the Stein structure on X. In the absence of topological obstructions the holomorphic map may be chosen to have pointwise maximal rank. The analogous result holds for any compact Hausdorff family of maps, but it fails in general for a noncompact family. Our main results are actually proved for smooth almost complex source manifolds (X,J) with the correct handlebody structure. The paper contains another proof of Eliashberg's (Int J Math 1:29--46, 1990) homotopy characterization of Stein manifolds and a slightly different explanation of the construction of exotic Stein surfaces due to Gompf (Ann Math 148 (2):619--693, 1998; J Symplectic Geom 3:565--587, 2005). (See also the related preprint math/0509419).Comment: The original publication is available at http://www.springerlink.co

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A new class of polynomially convex sets

Cited by 48 publications

References 12 publications

Totally real discs in non-pseudoconvex boundaries

Totally real discs in non-pseudoconvex boundaries

Polynomial Approximation, Local Polynomial Convexity, and Degenerate Cr Singularities — Ii

Stein structures and holomorphic mappings

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