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Brody curves omitting hyperplanes. (English) Zbl 1242.30019
The author considers holomorphic curves $$f : \mathbb{C} \to \mathbb{P}^n$$, where $$\mathbb{P}^n$$ denotes the complex projective space. The spherical derivative of $$f$$ is defined by $\|f'\| = \|f\|^{-4} \sum_{i<j} |f_{i}' f_{j} - f_{i} f_{j}'|^2$ where $$(f_0,\dots,f_n)$$ is a homogeneous representation of $$f$$ (i.e. $$f_j$$ are entire functions which never vanish simultaneously), and $\|f\|^2 = \sum_{j=0}^n |f_j|^2\,.$ If the spherical derivative is bounded, then $$f$$ is called a Brody curve. The Nevanlinna characteristic of $$f$$ is defined by $T(r,f) = \int_0^r \left( {1 \over \pi} \int_{|z| \leq t} \|f'\|^2(z)\,dm_z \right)\,{dt \over t}\,,$ where $$dm$$ is the area element in $$\mathbb{C}$$. Therefore, Brody curves are of order at most two and normal type, that is $T(r,f) = O(r^2)\,.$ In this paper the author proves that if a Brody curve $$f$$ omits $$n$$ hyperplanes in general position, then $T(r,f) = O(r)\,,$ which means that $$f$$ is of order at most one and normal type. This generalizes a result of J. Clunie and W. K. Hayman [Comment. Math. Helv. 40, 117–148 (1966; Zbl 0142.04303)] who proved this in the special case $$n=1$$. A different proof for $$n=1$$ is due to C. Pommerenke [Ann. Acad. Sci. Fenn., Ser. A I 476 (1970; Zbl 0205.09002)].

##### MSC:
 30D15 Special classes of entire functions of one complex variable and growth estimates 32Q99 Complex manifolds 32H30 Value distribution theory in higher dimensions
##### Keywords:
holomorphic curve; spherical derivative; order of growth
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