On Picard's Problem via Nevanlinna Theory II

Abstract

This work continues the author's earlier work (2026, Studia Mathematica) on Picard's problem: is every meromorphic function on a complete noncompact K\"ahler manifold with nonnegative Ricci curvature necessarily a constant, if it avoids 3 distinct values? In that prior work, a positive answer was obtained under a growth condition for non-parabolic manifolds. In this paper, we give a full solution to the non-parabolic case by removing this growth condition via a global Green function approach. For the parabolic case, to overcome the obstacle arising from the absence of a positive global Green function, we introduce a heat kernel approach to Nevanlinna theory. Based on it, we develop a Carlson-Griffiths theory, which gives the first systematic result in Nevanlinna theory for parabolic K\"ahler manifolds. As a direct application, we confirm the parabolic case of Picard's problem under a weak growth condition.