On Picard's Problem via Nevanlinna Theory II

TL;DR

This paper advances the understanding of Picard's problem on K"ahler manifolds by removing growth conditions in the non-parabolic case and introducing a heat kernel approach for the parabolic case, leading to new systematic results.

Contribution

It provides a full solution to Picard's problem on noncompact K"ahler manifolds by developing a Green function approach for non-parabolic cases and a heat kernel method for parabolic cases, establishing a systematic Nevanlinna theory.

Findings

Confirmed Picard's problem for non-parabolic manifolds without growth conditions.

Developed a systematic Nevanlinna theory for parabolic K"ahler manifolds.

Introduced a heat kernel approach to Nevanlinna theory for the first time in this context.

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.