Asymptotic Behavior of Normal Mappings of Several Complex Variables
Let M and N be connected Hermitian manifolds of dimensions m and n with Hermitian metrics hM and hN, respectively. Then the space ℓ(M, N) of continuous mappings between M and N endowed with the compact-open topology is second countable so that a metric can be furnished in ℓ(M, N) which induces the c...
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| Published in: | Canadian journal of mathematics Vol. 36; no. 4; pp. 718 - 746 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Cambridge, UK
Cambridge University Press
01.08.1984
|
| ISSN: | 0008-414X, 1496-4279 |
| Online Access: | Get full text |
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| Abstract | Let M and N be connected Hermitian manifolds of dimensions m and n with Hermitian metrics hM
and hN, respectively. Then the space ℓ(M, N) of continuous mappings between M and N endowed with the compact-open topology is second countable so that a metric can be furnished in ℓ(M, N) which induces the compact-open topology. A sequence {fn} in ℓ(M, N) converges to a n f in ℓ(M, N) in this topology if and only if fn
converges to f uniformly on compact subsets of M. It is then an easy consequence of the Cauchy integral formula to show that the space ℋ(M, N) of holomorphic mappings f:M → N is a closed subspace of ℓ(M, N). In this paper, generalizing the classical notions of normal functions, Bloch functions, regular sequences and P-point sequences of one complex variable to the mappings in ℋ(M, N), see also [25], we obtain various relations which exist between these notions. |
|---|---|
| AbstractList | Let
M
and
N
be connected Hermitian manifolds of dimensions
m
and
n
with Hermitian metrics
h
M
and
h
N
,
respectively. Then the space
ℓ
(
M, N
) of continuous mappings between
M
and
N
endowed with the compact-open topology is second countable so that a metric can be furnished in
ℓ
(
M, N
) which induces the compact-open topology. A sequence
{f
n
}
in ℓ(
M, N
) converges to a n
f
in ℓ(
M, N
) in this topology if and only if
f
n
converges to
f
uniformly on compact subsets of
M.
It is then an easy consequence of the Cauchy integral formula to show that the space ℋ(
M, N
) of holomorphic mappings
f
:
M
→
N
is a closed subspace of
ℓ
(
M, N
).
In this paper, generalizing the classical notions of normal functions, Bloch functions, regular sequences and P-point sequences of one complex variable to the mappings in
ℋ
(
M, N
), see also [
25
], we obtain various relations which exist between these notions. Let M and N be connected Hermitian manifolds of dimensions m and n with Hermitian metrics hM and hN, respectively. Then the space ℓ(M, N) of continuous mappings between M and N endowed with the compact-open topology is second countable so that a metric can be furnished in ℓ(M, N) which induces the compact-open topology. A sequence {fn} in ℓ(M, N) converges to a n f in ℓ(M, N) in this topology if and only if fn converges to f uniformly on compact subsets of M. It is then an easy consequence of the Cauchy integral formula to show that the space ℋ(M, N) of holomorphic mappings f:M → N is a closed subspace of ℓ(M, N). In this paper, generalizing the classical notions of normal functions, Bloch functions, regular sequences and P-point sequences of one complex variable to the mappings in ℋ(M, N), see also [25], we obtain various relations which exist between these notions. |
| Author | Hahn, Kyong T. |
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| Copyright | Copyright © Canadian Mathematical Society 1984 |
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| References | Hahn (S0008414X00021775_R0526) 1978; 30 Gauthier (S0008414X00021775_R0520) 1968; 32 Wu (S0008414X00021775_R0538) 1967; 119 Cima (S0008414X00021775_R0516) 1983; 50 Lohwater (S0008414X00021775_R0531) 1973; 550 Barth (S0008414X00021775_R0514) 1970; 24 Cirka (S0008414X00021775_R0517) 1973; 92 Seidel (S0008414X00021775_R0533) 1942; 52 Gavrilov (S0008414X00021775_R0522) 1982; 26 Brody (S0008414X00021775_R0515) 1978; 235 Dovbus (S0008414X00021775_R0519) 1982; 25 Hahn (S0008414X00021775_R0525) 1977; 68 Hahn (S0008414X00021775_R0524) 1975; 27 Kiernan (S0008414X00021775_R0528) 1970; 76 Gavrilov (S0008414X00021775_R0521) 1965; 109 Lehto (S0008414X00021775_R0530) 1957; 97 Diederieh (S0008414X00021775_R0518) 1970; 187 Timoney (S0008414X00021775_R0535) 1978 Stein (S0008414X00021775_R0534) 1972 Anderson (S0008414X00021775_R0513) 1974; 270 S0008414X00021775_R0537 Wicker (S0008414X00021775_R0536) 1977; 29 Graham (S0008414X00021775_R0523) 1975; 207 Kerzman (S0008414X00021775_R0527) 1981; 257 Kobayashi (S0008414X00021775_R0529) 1970 Royden (S0008414X00021775_R0532) 1971; 185 |
| References_xml | – volume: 26 start-page: 186 year: 1982 ident: S0008414X00021775_R0522 article-title: Soviet Math. Dokl. publication-title: Boundary singularities generated by cluster sets of functions of several complex variables, – volume: 270 start-page: 12 year: 1974 ident: S0008414X00021775_R0513 article-title: J. Reine Angew. Math. publication-title: On Block functions and normal Junctions, – volume-title: Boundary behavior of holomorphic functions of several complex variables year: 1972 ident: S0008414X00021775_R0534 – volume: 52 start-page: 128 year: 1942 ident: S0008414X00021775_R0533 article-title: Trans. Amer. Math. Soc. publication-title: On the derivatives of functions analytic in the unit circle and their radii of univalence and of p-valence. – volume: 27 start-page: 446 year: 1975 ident: S0008414X00021775_R0524 article-title: Can. J. Math. publication-title: Holomorphic mappings of the hyperbolic space into the complex euclidean space and Bloch theorem, – volume: 550 year: 1973 ident: S0008414X00021775_R0531 article-title: Ann. Acad. Sci. Fenn., Ser. A I publication-title: On normal meromorphic functions, – volume: 257 start-page: 171 year: 1981 ident: S0008414X00021775_R0527 article-title: Math. Ann. publication-title: Fonctions pluri-sousharmoniques d'exhaustion bornées et domaines taut, – volume-title: Hyperbolic manifolds and holomorphic mappings year: 1970 ident: S0008414X00021775_R0529 – volume: 92 start-page: 622 year: 1973 ident: S0008414X00021775_R0517 article-title: Mat. Sb. publication-title: The theorems of Lindelöf and Fatou in Cn – volume: 29 start-page: 299 year: 1977 ident: S0008414X00021775_R0536 article-title: Can. J. Math. publication-title: Generalized Block mappings in complex Hilbert space. – volume: 207 start-page: 219 year: 1975 ident: S0008414X00021775_R0523 article-title: Trans. Amer. Math. Soc. publication-title: Boundary behavior of the Caratheodory and Kobayashi metrics on strongly pseudoconvex domains in Cn with smooth boundary, – volume: 25 start-page: 267 year: 1982 ident: S0008414X00021775_R0519 article-title: Soviet Math. Dokl. publication-title: Boundary behavior of normal holomorphic junctions of several complex variables, – volume: 97 start-page: 47 year: 1957 ident: S0008414X00021775_R0530 article-title: Acta. Math. publication-title: Boundary behavior and normal meromorphic functions. – volume: 235 start-page: 213 year: 1978 ident: S0008414X00021775_R0515 article-title: Trans. Amer. Math. Soc. publication-title: Compact manifolds and hyperbolicity, – volume: 119 start-page: 193 year: 1967 ident: S0008414X00021775_R0538 article-title: Acta Math. publication-title: Normal families of holomorphic mappings. – volume: 24 start-page: 429 year: 1970 ident: S0008414X00021775_R0514 article-title: Proc. Amer. Math. Soc. publication-title: Taut and tight complex manifolds, – volume: 109 start-page: 408 year: 1965 ident: S0008414X00021775_R0521 article-title: Mat. Sb. publication-title: On the distribution of values of non-normal meromorphic junctions in the unit disc (Russian), – volume-title: Bloch functions in several complex variables. year: 1978 ident: S0008414X00021775_R0535 – volume: 187 start-page: 9 year: 1970 ident: S0008414X00021775_R0518 article-title: Math. Ann. publication-title: Das Randverhalton der Bergmansehen Kernjunktion und Metrik in Streng pseudo-komvexen Gebieten, – volume: 50 start-page: 303 year: 1983 ident: S0008414X00021775_R0516 article-title: Duke Math. J. publication-title: Lindelöf principle and normal junctions of several complex variables, – volume: 30 start-page: 22 year: 1978 ident: S0008414X00021775_R0526 article-title: Can. J. Math. publication-title: Geometry of the unit ball of a complex Hilbert space, – volume: 68 start-page: 437 year: 1977 ident: S0008414X00021775_R0525 article-title: Pacific J. Math. publication-title: On completeness of the Bergman metric and its subordinate metrics, II, – ident: S0008414X00021775_R0537 – volume: 32 start-page: 277 year: 1968 ident: S0008414X00021775_R0520 article-title: Nagoya Math. J. publication-title: A criterion for normal, – volume: 185 start-page: 125 year: 1971 ident: S0008414X00021775_R0532 article-title: Lecture Notes in Math. publication-title: Remarks on the Kobayashi metric, several complex variables II, – volume: 76 start-page: 49 year: 1970 ident: S0008414X00021775_R0528 article-title: Bull. Amer. Math. Soc. publication-title: On the relations between taut, tight and hyperbolic mamjolds. |
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and hN, respectively. Then the space ℓ(M, N) of continuous... Let M and N be connected Hermitian manifolds of dimensions m and n with Hermitian metrics h M and h N , respectively. Then the space ℓ ( M, N ) of continuous... |
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| Title | Asymptotic Behavior of Normal Mappings of Several Complex Variables |
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