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

TL;DR

This paper proves a Picard-type theorem for entire curves in complex projective space with vanishing spherical derivative on inverse images of hypersurfaces, leading to conditions under which such curves are Brody curves.

Contribution

It introduces a new Picard-type theorem for entire curves in projective space based on the behavior of their spherical derivatives relative to hypersurfaces.

Findings

Established a Picard-type theorem for entire curves with vanishing spherical derivative.

Identified a finite union of hypersurfaces where bounded spherical derivative implies Brody curve.

Connected the boundedness of spherical derivative on inverse images to the curve being Brody.

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.