On the number of asymptotic points of holomorphic curves (Q1387866)

The author generalizes Ahlfors' theorem on the number of asymptotic values of meromorphic functions to holomorphic curves. Let \(f=[f_1, \dots,f_{n+1}]\) be a transcendental holomorphic curve \(f:\mathbb{C}\to\mathbb{P}^n\mathbb{C}\) with the reduced representation \((f_1,\dots, f_{n+1}): \mathbb{C}\to \mathbb{C}^{n+1} \setminus \{0\}\), \(n>0\). Main Theorem established: If \(\lambda\), the lower order of \(f\), is finite, then the number \(N\) of asymptotic spots of \(f\), which are of first kind and distinct in general position, verifies the inequalities \[ N\leq n\text{ if }\lambda\leq 1/2n,\;N\leq 2n-1\text{ if }1/2n<\lambda<1 \text{ and }N\leq 2n\lambda \text{ if }1\leq \lambda< \infty. \] Definitions and notations: For \(a\in\mathbb{C}^{n+1}\), \(d(a,f(z))=|(a,f(z))|/ \| A\| \| f(z) \|\) (as in \textit{H. Weyl} `Meromorphic functions and analytic curves', Princeton Univ. Press (1943; Zbl 0061.15302)); \[ \lambda= \liminf_{r \to \infty} \bigl(\log T(r,f)/ \log r\bigr); \] \[ T(r,f)=(1/2\pi) \int^{2\pi}_0 \log\bigl\| f(re^{i\theta})\bigr\| d\theta-\log\bigl\| f(0)\bigr \|. \] Let \(V=\{a\in\mathbb{C}^{n+1}: (a,f)=0\}\). A point \(a\in\mathbb{C}^{n+1} \setminus V\) is an asymptotic point of \(f\) iff there is a path \(\Gamma:z= z(t)\), \(0\leq t<1\), in \(\mathbb{C}\) such that: \(\lim_{t\to 1}z(t)=\infty\) and \(\lim_{t\to 1}d(a,f(z(t))) =0\). Let \(a\in\mathbb{C}^{n+1}\setminus V\) such that for any \(x\in(0,1]\), the open set \(D(x;a)=\{z:d(a,f(z))<x\}\) has at least one not relatively compact component. An asymptotic spot \(\sigma(x;a)\) of \(f\) corresponding to \(a\) is a function defined on \((0,1]\) and satisfying: i) for each \(x\in(0,1]\), \(\sigma (x; a)\) is a component of \(D(x;a)\) which is not relatively compact; ii) if \(x_1 <x_2 \), then \(\sigma(x_1;a)\subset\sigma (x_2;a)\). Two asymptotic spots \(\sigma_1 (x; a)\) and \(\sigma_2(x;b)\) are distinct iff either \(a\neq b\) or \(a=b\) but there exists an \(x\in(0,1]\) such that \(\sigma_1(x;a) \cap\sigma_2(x;a) =\emptyset\). The elements of a set \(S_f\) of asymptotic spots of \(f\) are distinct in general position if they are distinct and if \(A(S_f)=\{a: \sigma(x;a)\in S_f\}\) is in general position, (i.e. if the elements of any subset of \(A(S_f)\), whose cardinal is \(\leq n+1\), are linearly independent). An asymptotic spot \(\sigma (x;a)\) of \(f\) is of the first kind if there exists \(\delta\in(0,1)\) such that for any \(x\in(0,\delta)\), \(\sigma(x;a)\) does not contain zeros of \((a,f)\). Evidently, \(a\) is an asymptotic point of \(f\) iff there exists an asymptotic spot of \(f\) corresponding to \(a\). The proof of the theorem is based on interesting lemmas concerning properties of the asymptotic spots. Examples illustrate the results.