分担超平面的全纯曲线族的正规定则

Normal Criteria Concerning Shared Hyperplanes for Families of Holomorphic Curves

摘要: 本文利用值分布理论和正规族理论等相关知识，研究了全纯曲线族分担处于次一般位置的超平面的正规定则。设ℱ是一族从区域D⊂ℂ到ℙ3(ℂ)的全纯曲线，Hl={x∈P3(C):〈x,αl〉=0}≠H0是ℙ3(ℂ)中k个处于t次一般位置的超平面，其中αl=(αl0,αl1,αl2,αl3)T，l=1,2,⋯,k，H0={x0=0}，t≥3，k=min{p:2t1≤p≤3t1,[p−t3]≤(p−2t−1)}。如果对任意的f∈ℱ，满足：f(z)∈Hl当且仅当∇f(z)∈Hl；若f(z)∈∪t=1kHl，那么|〈f(z),H0〉|||f(z)|⋅|H0||≥δ，其中0<δ<1是一个常数，则ℱ在D上正规。

Abstract: Based on value distribution theory and normal family theory, the normality of hyperplanes in sub-general position shared by holomorphic curve families is considered. Letℱbe a family of holomorphic maps of a domainD⊂ℂtoℙ3(ℂ). LetHl={x∈P3(C):〈x,αl〉=0}≠H0be hyperplanes inℙ3(ℂ)located in general position, whereαl=(αl0,αl1,αl2,αl3)T,l=1,2,⋯,k,H0={x0=0},k=min{p:2t1≤p≤3t1,[p−t3]≤(p−2t−1)}. Assume the following conditions hold for everyf∈ℱ: if and only iff(z)∈Hl, then∇f(z)∈Hl; Iff(z)∈∪t=1kHl, then|〈f(z),H0〉|||f(z)|⋅|H0||≥δ, where0<δ<1is a constant. Thenℱis normal on D.