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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
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