Holomorphic approximation on totally real submanifolds of a complex manifold

Harvey, F. Reese; Wells, Raymond O.

doi:10.1090/s0002-9904-1971-12820-3

Cited by 6 publications

(5 citation statements)

References 7 publications

(9 reference statements)

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“…First, we apply the above result (or an earlier version by Harvey-Wells; see [19]) to X D and X 0 D ; to obtain that any f 2 C./ can be approximated uniformly on by functions holomorphic on a neighborhood of . Since is polynomially convex, the Oka-Weil theorem allows us to conclude that C ./ D P ./.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Polynomially convex embeddings of odd-dimensional closed manifolds

Gupta

Shafikov

2021

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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It is shown that any smooth closed orientable manifold of dimension 2 ⁢ k + 1 {2k+1} , k ≥ 2 {k\geq 2} , admits a smooth polynomially convex embedding into ℂ 3 ⁢ k {\mathbb{C}^{3k}} . This improves by 1 the previously known lower bound of 3 ⁢ k + 1 {3k+1} on the possible ambient complex dimension for such embeddings (which is sharp when k = 1 {k=1} ). It is further shown that the embeddings produced have the property that all continuous functions on the image can be uniformly approximated by holomorphic polynomials. Lastly, the same technique is modified to construct embeddings whose images have nontrivial hulls containing no nontrivial analytic disks. The distinguishing feature of this dimensional setting is the appearance of nonisolated CR-singularities, which cannot be tackled using only local analytic methods (as done in earlier results of this kind), and a topological approach is required.

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Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Polynomially convex embeddings of odd-dimensional closed manifolds

Gupta

Shafikov

2021

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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“…This is a C 2 vector field on ∂G. Approximate n on the compact set γ ([−ε, 1]) by an analytic vector field N in a neighbourhood (in C 2 ) of this set ( [8]). Put H(z, t + is) = F N ,s (H(z, t)), s ∈ [0, σ ] for a small positive number σ .…”

Section: It Remains To Prove the Lemmamentioning

confidence: 99%

On propagation of boundary continuity of holomorphic functions of several variables

Franzén¹,

Jöricke

2009

Annali Scuola Normale Superiore - Classe Di Scienze

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We prove that continuity properties of bounded analytic functions in bounded smoothly bounded pseudoconvex domains in two-dimensional affine space are determined by their behaviour near the Shilov boundary. Namely, if the function has continuous extension to an open subset of the boundary containing the Shilov boundary it extends continuously to the whole boundary. If it is e.g. Hölder continuous on such a boundary set, it is Hölder continuous on the closure of the domain. The statements may fail if the boundary is not smooth.

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“…A compact set K is said to be holomorphically convex if K and the spectrum of A{K) are homeomorphic under the natural map. In [5] a notion of the "envelope of holomorphy", for K a compact subset of C n , was introduced; there it was proved, in particular, that K is equal to its envelope if and only if K is holomorphically convex. The Cartan Theorems A and B for open holomorphically convex sets in C n admit analogues for compact holomorphically convex sets in C n (see [5]).…”

Section: Michael Freeman and Reese Harveymentioning

confidence: 99%

“…Thus T is the envelope of holomorphy of T (see [5] for the precise definition of envelope of holomorphy of T).…”

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confidence: 99%