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.