Some classical theorems and families of normal maps in several complex variables

Joseph, James E.; Kwack, Myung H.

doi:10.1080/17476939608814902

Cited by 17 publications

(25 citation statements)

References 16 publications

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“…Proof. As for the proofs of (1) and (2) [11]. To finish the proof of (1), if f ∈ F, then there exists a sequence {f n } in F such that f n → f .…”

Section: Extension and Convergence Theoremsmentioning

confidence: 99%

“…There exists r, 0 < r < 1, such that ultimately g n (D * r ) ⊂ V ; so ultimately g n extends to g n ∈ H(D, Y ). From Theorem 2 in [11] there exists a subsequence of …”

Section: Extension and Convergence Theoremsmentioning

confidence: 99%

“…Abate [1] has shown that a complex space X is hyperbolic iff H(D, X) is relatively compact in C(D, X + ). After preliminaries in §1, we offer our main results in §2, providing generalizations of Picard extension theorems from [11], [14], [15], [16], [19], of the MontelCarathéodory Theorem and of a Noguchi extension-convergence theorem (see [5, p. 300], [22, pp.…”

Section: Introductionmentioning

confidence: 99%

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Extension and convergence theorems for families of normal maps in several complex variables

Joseph

Kwack

1997

Proc. Amer. Math. Soc.

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Abstract. Let H(X, Y ) ( C(X, Y ) ) represent the family of holomorphic (continuous) maps from a complex (topological) space X to a complex (topological) space Y , and let(with the compact-open topology) for each complex manifold M . We show that normal maps as defined and studied by authors in various settings are, as singleton sets, uniformly normal families, and prove extension and convergence theorems for uniformly normal families. These theorems include (1) extension theorems of big Picard type for such families -defined on complex manifolds having divisors with normal crossings -which encompass results of Järvi, Kiernan, Kobayashi, and Kwack as special cases, and (2) generalizations to such families of an extension-convergence theorem due to Noguchi.

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“…Proof. As for the proofs of (1) and (2) [11]. To finish the proof of (1), if f ∈ F, then there exists a sequence {f n } in F such that f n → f .…”

Section: Extension and Convergence Theoremsmentioning

confidence: 99%

“…There exists r, 0 < r < 1, such that ultimately g n (D * r ) ⊂ V ; so ultimately g n extends to g n ∈ H(D, Y ). From Theorem 2 in [11] there exists a subsequence of …”

Section: Extension and Convergence Theoremsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Extension and convergence theorems for families of normal maps in several complex variables

Joseph

Kwack

1997

Proc. Amer. Math. Soc.

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“…By [JK1,Theorem 2.3], f extends to a holomorphic mapping f * : ∆ n → M . Then every f ∈ F extends to a holomorphic mapping f * : ∆ n → M and the family F * := {f * : f ∈ F} ⊂ Hol(∆ n , M ) is uniformly normal.…”

Section: Examplesmentioning

confidence: 99%

“…Since then normal holomorphic maps has been studied intensively, resulting in an extensive development in the single complex variable context and in generalizations to several complex variables settings (see [Za], [JK1], [JK2], [AK] and the references cited in [Za] and [JK2]). …”

Section: §1 Introductionmentioning

confidence: 99%

Normal Families of meromorphic mappings of several complex variables into P^N(C)

Ngoc

Thai²,

Trang

2005

Nagoya Mathematical Journal

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Abstract. The first aim in this article is to give some sufficient conditions for a family of meromorphic mappings of a domain D in C n into P N (C) omitting hypersurfaces to be meromorphically normal. Our result is a generalization of the results of Fujimoto and Tu. The second aim is to investigate extending holomorphic mappings into the compact complex space from the viewpoint of the theory of meromorphically normal families of meromorphic mappings. §1. Introduction Classically, a family F of holomorphic functions on a domain D ⊂ C is said to be (holomorphically) normal if every sequence in F contains a subsequence which converges uniformly on all the compact subsets of D.In 1957 Lehto and Virtanen [LeVi] introduced the concept of normal meromorphic functions in connection with the study of boundary behaviour of meromorphic functions of one complex variable. Since then normal holomorphic maps has been studied intensively, resulting in an extensive development The first aim in this article is to give some sufficient conditions for a family of meromorphic mappings of a domain D in C n into P N (C) omitting hypersurfaces to be meromorphically normal or quasi-normal. These results are generalizations of the above Fujimoto's and Tu's results.The second aim of this article is to investigate extending holomorphic mappings into compact complex spaces from the viewpoint of the theory of meromorphically normal families of meromorphic mappings. In order to state our main result, we need some preliminary. First, for hypersurfaces H i (1 ≤ i ≤ q) of P N (C) with q ≥ N + 1, let Q i (1 ≤ i ≤ q) be their defining polynomials, i.e., the homogeneous polynomials without multiple factors such thatHere and below, throughout the article, we only consider homogeneous polynomials Q(z) = a ν z ν normalized so that |a ν | 2 = 1. Now we definewhere z = |z j | 2 1/2 . Next, let Λ d (S) denote the real d-dimensional Hausdorff measure of S ⊂ C n . For a formal Z-linear combination X = i∈I n i X i of analytic subsets X i ⊂ C n and for a subset E ⊂ C n , we call i∈I Λ d (X i ∩ E) (resp. i∈I n i Λ d (X i ∩ E)), the d-dimensional Lebesgue area of X ∩ E regardless of multiplicities (resp. with counting multiplicities).Now we can state our main results.

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Applications of Zalcman’s Lemma in C N

Dovbush

2021

Trends in Mathematics

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Some classical theorems and families of normal maps in several complex variables

Cited by 17 publications

References 16 publications

Extension and convergence theorems for families of normal maps in several complex variables

Extension and convergence theorems for families of normal maps in several complex variables

Normal Families of meromorphic mappings of several complex variables into P^N(C)

Applications of Zalcman’s Lemma in C N

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Some classical theorems and families of normal maps in several complex variables

Cited by 17 publications

References 16 publications

Extension and convergence theorems for families of normal maps in several complex variables

Extension and convergence theorems for families of normal maps in several complex variables

Normal Families of meromorphic mappings of several complex variables into PN(C)

Applications of Zalcman’s Lemma in C N

Contact Info

Product

Resources

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Normal Families of meromorphic mappings of several complex variables into P^N(C)