Normal families of meromorphic mappings of several complex variables into PN (C) for moving targets

Montel-type theorems in several complex variables with continuously moving targets

“…For example (see, e.g., [18, p. 40]), f (z 1 , z 2 ) := (1, e z ) (from C to C 2 ) omits three moving hyperplanes Z 0 = 0, Z 1 = 0, e z Z 0 +Z 1 = 0 (z ∈ C, (Z 0 , Z 1 ) ∈ C 2 ), where these three moving hyperplanes are located in pointwise general position on C. However, f (z) := ρ( f (z)) (from C to P 1 (C)) is not a constant. Although a holomorphic mapping from C into P N (C) omitting 2N + 1 moving hyperplanes in pointwise general position may not be a constant, Tu and Li [17] extended Theorem 2.2 to the case of moving hyperplanes in pointwise general position and proved the following results. Theorem 2.3 (see [17]) Let F be a family of holomorphic mappings of a domain D in C n into P N (C), and H 1 (z), · · · , H 2N +1 (z) (z ∈ D) be 2N + 1 moving hyperplanes in P N (C) located in pointwise general position on D. If each f in F omits H j (z) (j = 1, · · · , 2N + 1) on D, then F is a normal family on D.…”

“…Although a holomorphic mapping from C into P N (C) omitting 2N + 1 moving hyperplanes in pointwise general position may not be a constant, Tu and Li [17] extended Theorem 2.2 to the case of moving hyperplanes in pointwise general position and proved the following results. Theorem 2.3 (see [17]) Let F be a family of holomorphic mappings of a domain D in C n into P N (C), and H 1 (z), · · · , H 2N +1 (z) (z ∈ D) be 2N + 1 moving hyperplanes in P N (C) located in pointwise general position on D. If each f in F omits H j (z) (j = 1, · · · , 2N + 1) on D, then F is a normal family on D.…”

“…Since a holomorphic mapping from C into P N (C) omitting 2N + 1 moving hyperplanes in pointwise general position may not be a constant, there are some gaps for one to use the idea in proving the normality with moving targets. In the proof of Theorem 2.3 (with moving hyperplanes), the key idea (see [17]) is as follows: when one applies Zaclman's lemma to produce a nonconstant holomorphic map h from C into P N (C), h can actually omit a set of fixed hyperplanes by Hurwitz's theorem. Hence, the Picard-type theorems with fixed hyperplanes could be applied to get the normality with moving hyperplanes.…”

“…orem of holomorphic mappings of C into P N (C) to the case of moving hyperplanes in pointwise general position. By starting from Fujimoto-Green's and Nochka's Picard-type theorems and using the heuristic principle obtained by Aladro and Krantz [2], Tu and Li [17] obtained some normality criteria for families of holomorphic mappings of several complex variables into P N (C) for moving hyperplanes in pointwise general position. Bargmann, Bonk, Hinkkanen and Martin [3] introduced a new idea related to Montel's theorem and proved a normality critrion for families of meromorphic functions omitting three continuous functions.…”

Normal families of meromorphic mappings of several complex variables into CPnfor moving hypersurfaces

“…In 2005, Tu and Li [13] extended the above theorem to the case of moving hyperplanes as follows: Theorem 1.2. Let F be a family of holomorphic mappings of a domain D ⊂ C m into CP n and let {H j } q j=1 be q (≥ 2n + 1) moving hyperplanes in CP n in pointwise general position on D such that each f in F intersects H j on D with multiplicity at least m j (j = 1, .…”