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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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


Authors: James E. Joseph and Myung H. Kwack
Journal: Proc. Amer. Math. Soc. 125 (1997), 1675-1684
MSC (1991): Primary 32A10, 32C10, 32H20, 32A17; Secondary 54C20, 54C35, 54D35, 54C05
MathSciNet review: 1423310
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $\mathcal {H}(X,Y)\ (\ \mathcal {C}(X,Y)\ )$ represent the family of holomorphic (continuous) maps from a complex (topological) space $X$ to a complex (topological) space $Y$, and let $Y^{+} = Y \cup \{\infty \}$ be the Alexandroff one-point compactification of $Y$ if $Y$ is not compact, $Y^{+}=Y$ if $Y$ is compact. We say that $\mathcal {F} \subset \mathcal {H}(X,Y)$ is uniformly normal if $\{f \circ \varphi : f \in \mathcal {F}$, $\varphi \in \mathcal {H}(M,X)\}$ is relatively compact in $\mathcal {C}(M,Y^{+})$ (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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Additional Information

James E. Joseph
Affiliation: Department of Mathematics, Howard University, Washington, D. C. 20059
Email: jjoseph@scs.howard.edu

Myung H. Kwack
Affiliation: Department of Mathematics, Howard University, Washington, D. C. 20059

DOI: http://dx.doi.org/10.1090/S0002-9939-97-04117-8
PII: S 0002-9939(97)04117-8
Keywords: Holomorphic maps, normal maps, uniformly normal families, complex spaces, length function, hyperbolic complex manifolds, hyperbolically imbedded, function spaces, continuous extensions, Picard Theorem, compact--open topology, Ascoli--Arzel\`{a} Theorem, one--point compactification
Received by editor(s): June 8, 1995
Dedicated: Dedicated to Professor Shoshichi Kobayashi at his retirement
Communicated by: Eric Bedford
Article copyright: © Copyright 1997 American Mathematical Society