Montel-type theorems in several complex variables with continuously moving targets
Let \(P^N(\mathbb C)\) be the complex projective space of dimension \(N,\) and \( \rho: \mathbb C^{N+1}\to P^N(\mathbb C)\) be the standard projective mapping. Let \(h\) be a holomorphic mapping from a domain \(D\) in \(\mathbb C^n\) into \(P^N(\mathbb C)\). For any \(a \in D\), the mapping \(h\) has a reduced representation \(\widetilde{h}^{(p)}(z)=(h^{(p)}_0(z), \dots, h^{(p)}_N(z))\) on some neighborhood \(U\) of \(a\) in \(D,\) i.e., \(h^{(p)}_0(z), \dots, h^{(p)}(z)\) are holomorphic on \(U\) without common zeros such that \(h(z)=\rho(\widetilde{h}(z))\) on \(U\). A family \(F\) of holomorphic mappings of a domain \(D\) in \(\mathbb C^n\) into \(P^N(\mathbb C)\) is said to be normal on \(D\) if any sequence in \(F\) contains a subsequence which converges uniformly on compact subsets of \(D\) to a holomorphic mapping of \(D\) into \(P^N(\mathbb C),\) where a subsequence \(\{f^p\}\subset F\) is said to converge uniformly on compact subsets of \(D\) to a holomorphic mapping \(f\) of \(D\) into \(P^N(\mathbb C)\) iff, for any \(a \in D,\) each \(f^{(p)}(z)\) has a reduced representation \(\widetilde{f}^{(p)}(z)=(f^{(p)}_0(z), \dots, f^{(p)}_N(z))\) on some fixed neighborhood \(U\) of \(a\in D\) such that \(\{f^{(p)}_i(z)\}_{i=1}^\infty\) converge uniformly on compact subsets of \(U\) to a holomorphic function \(f_i(z)\) \((i=0, \dots, N)\) on \(D\) with the property that \(\widetilde{f}(z):=(f_0(z), \ldots, f_N(z))\) is a reduced representation of \(f(z)\) on \(U\). Let \(E\) be a subset of \(\mathbb C^n\). For \(z\in E\), define the linear form \[ L(z)(Z_0,\ldots,Z_N):=\sum_{j=0}^Na_j(z)Z_j, \] where \(a_j(z)\) \((j=0,\ldots,N)\) are continuous functions on \(E\) without common zeros. Then \[ H(z):= \rho(\{(Z_0,\ldots,Z_N)\in\mathbb C^{N+1}\setminus \{0\} : L(z)(Z_0,\ldots,Z_N)=0\}) \] is called a continuous moving hyperplane in \(P^N(\mathbb C)\) corresponding to the linear form \(L(z)\) \((z \in E)\). Continuous moving hyperplanes in \(H_j(z)\) (where \(j=1,\ldots, q: q\geq N+1\)) are said to be located in \(P^N(\mathbb C)\) in pointwise general position on \(E\) if, for any fixed point \(z_0 \in E\), the hyperplanes \(H_j(z_0)\) \((j=1,\ldots, q: q\geq N+1)\) are located in general position. When \(E\) is a domain in \(\mathbb C^n\) and \(a_j(z)\) \((0\leq j \leq N)\) are holomorphic in \(E,\) \(H(z)\) is said to be a moving hyperplane in \(P^N(\mathbb C)\). Set \(F(z):=L(z)(f_0(z), \dots, f_N(z))\). When \(f(z)\in P^N(\mathbb C)\setminus H(z)\) for all \(z \in D\), we say that \(f\) omits the continuous moving hyperplane \(H(z)\) on \(D.\) Theorem 1. Let \(F\) be a family of holomorphic mappings of a domain \(D\) in \(\mathbb C^n\) into \(P^N(\mathbb C)\), and \(H_1(z),\dots,H_{2N+1}(z)\) \((z \in D)\) be \(2N+1\) continuously moving hyperplanes in \(P^N(\mathbb C)\) located in pointwise general position on \(D\). If each \(f\) in \(F\) omits \(H_j(z)\) \((j=1,\ldots, 2N+1)\) on \(D,\) then \(F\) is a normal family on \(D\). Corollary. Let \(F\) be a family of holomorphic mappings of the ball \(B(R)=\{z \in\mathbb C^n : \|z\|<R\}\) into \(P^1(\mathbb C)\) \((0<R<\infty)\). Then \(F\) is a normal family on \(B(R)\) iff for each sequence \(\{f_i(z)\}_{i=1}^\infty\) in \(F\), there exist three continuously moving hyperplanes \(H_1(z), H_2(z), H_3(z)\) \((z \in B(R))\) in \(P^1(\mathbb C)\) located in pointwise general position on \(B(R)\), such that for each closet ball \(\overline{B(r)}=\{z \in\mathbb C^n : \|z\|\leq r\}\) \((0<r<R)\), infinitely many mappings in \(\{f_i(z)\}_{i=1}^\infty\) omit \(H_j(z)\) \((j=1,2,3)\) on \(\overline{B(r)}.\) Let \(\Omega \subset\mathbb C^n\) be a domain and \(M\) be a compact complex manifold with Hermitian metric \(ds^2_M\). A holomorphic mapping \(f(z)\) from \(\Omega\) into \(M\) is said to be a normal holomorphic mapping from \(\Omega\) into \(M\) iff there exists a constant \(c\) such that for all \(z \in D\) and all \(\xi \in T_z(\Omega)\), \[ ds^2_M(f(z), df(z)(\xi))<cK_\Omega(z,\xi), \] where \(df(z)\) is the mapping from \(T_z(\Omega)\) into \(T_{f(z)}(M)\) induced by \(f\) and \(K_\Omega\) denotes the Kobayashi metric for \(\Omega\). Theorem 2. Let \(f\) be a holomorphic mapping from a bounded domain \(D\) in \(\mathbb C^n\) into \(P^N(\mathbb C)\), and \(H_1(z),\dots, H_{2N+1}(z)\) \((z \in \overline{D})\) be \(2N+1\) continuously moving hyperplanes in \(P^N(\mathbb C)\) located in pointwise general position on \(\overline{D}\). If \(f\) omits \(H_j(z)\) \((j=1,\dots, 2N+1)\) on \(D\), then \(f\) is a normal holomorphic mapping from \(D\) into \(P^N(\mathbb C)\).
scientific article; zbMATH DE number 5816604
scientific article