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.
The main aim of this article is to give sufficient conditions for a family of meromorphic mappings of a domain D in ℂn into ℙN(ℂ) to be meromorphically normal if they satisfy only some very weak conditions with respect to moving hypersurfaces in ℙN(ℂ), namely, that their intersections with these moving hypersurfaces, which moreover may depend on the meromorphic maps, are in some sense uniform. Our results generalize and complete previous results in this area, especially the works of Fujimoto, Tu, Tu-Li, Mai-Thai-Trang, and the recent work of Quang-Tan.
A necessary and sufficient condition for tautness of locally taut domains in a weakly Brody hyperbolic complex space is given. Moreover, some results of Kobayashi and Gaussier are deduced as corollaries.
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