A property of the spherical derivative of an entire curve in complex projective space

Abstract

We establish a type of the Picard's theorem for entire curves in $P^n(\mathbb{C})$ whose spherical derivative vanishes on the inverse images of hypersurface targets. Then, as a corollary, we prove that there is an union $D$ of finite number of hypersurfaces in the complex projective space $P^n(\mathbb{C})$ such that for every entire curve $f$ in $P^n(\mathbb{C})$, if the spherical derivative $f^{\#}$ of $f$ is bounded on $f^{-1}(D)$, then $f^{\#}$ is bounded on the entire complex plane, and hence, $f$ is a Brody curve.