On Picard's Problem via Nevanlinna Theory

TL;DR

This paper extends Picard's theorem to certain non-parabolic Kähler manifolds with non-negative Ricci curvature using Nevanlinna theory, showing meromorphic functions' value-omission properties under volume growth conditions.

Contribution

It develops a Nevanlinna theory framework to solve Picard's problem on specific Kähler manifolds, establishing value-omission limits for meromorphic functions.

Findings

Meromorphic functions on these manifolds reduce to constants if they omit three values.

Functions of non-polynomial growth can omit at most two values.

The results depend on a volume growth condition of the manifold.

Abstract

We consider the classical Picard's problem for non-parabolic complete K\"ahler manifolds with non-negative Ricci curvature. Based on the global Green function approach, we give a positive answer to Picard's problem under certain condition by developing Nevanlinna theory. That is, we prove that every meromorphic function on such a manifold reduces to a constant if it omits three distinct values, provided that the manifold satisfies a volume growth condition; and prove that every meromorphic function of non-polynomial type growth on such a manifold can avoid 2 distinct values at most.