导曲线分担超平面的正规定则

A Criterion of Normality That the Derived Curve Shares the Hyperplane

摘要: 本文利用正规族理论等相关知识，研究了导曲线分担处于一般位置的超平面的正规定则，得到了如下结果：设ℱ是一族从区域D⊂ℂ到ℙN(ℂ)的全纯曲线，Hl={x∈PN(C):〈x,αl〉=0}≠H0是ℙN(ℂ)中处于一般位置的超平面，其中αl=(αl0,αl1,⋯,αlN)T，l=1,2,⋯,2N+1，H0={x0=0}。如果对任意的f∈ℱ，满足：若∇f(z)∈Hl，则f(z)∈Hl；若f(z)∈∪l=12N+1Hl，那么|f(z),H0|||f(z)|⋅|H0||≥δ；若f(z)∈∪l=12N+1Hl，则|〈∇f(z),Hl〉||f0(z)|2≤1δ，其中0<δ<1是一个常数，则ℱ在D上正规。

Abstract: In this paper, using the normal family theory and other relevant knowledge, we study a criterion of normality that the derived curve shares the hyperplane in the general position, and get the following result: letℱbe a family of holomorphic maps of a domainD⊂ℂtoℙN(ℂ). LetHl={x∈PN(C):〈x,αl〉=0}≠H0be hyperplanes inℙN(ℂ)located in general position, whereαl=(αl0,αl1,⋯,αlN)T,l=1,2,⋯,2N+1,H0={x0=0}. Assume the following conditions holdfor everyf∈ℱ: if∇f(z)∈Hl, thenf(z)∈Hl; iff(z)∈∪l=12N+1Hl, then|〈f(z),H0〉|||f(z)|⋅|H0||≥δ; iff(z)∈∪l=12N+1Hlthen|〈∇f(z),Hl〉||f0(z)|2≤1δ, where0<δ<1is a constant. Thenℱis normal on D.