Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Picard's Problem via Nevanlinna Theory II

Xianjing Dong

Abstract

This work continues the author's earlier work (2026, Studia Mathematica) on Picard's problem: is every meromorphic function on a complete noncompact Kähler 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 Kähler manifolds. As a direct application, we confirm the parabolic case of Picard's problem under a weak growth condition.

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 Kähler 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 Kähler manifolds. As a direct application, we confirm the parabolic case of Picard's problem under a weak growth condition.

Paper Structure

This paper contains 16 sections, 35 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Ricci Curvature
  4. Volume Criterion for Non-parabolicity
  5. Volume Comparison Theorem
  6. Heat Kernel
  7. Carlson-Griffiths Theory for Non-parabolic Complete Kähler Manifolds with Nonnegative Ricci Curvature
  8. Construction of $\Delta(r)$
  9. Nevanlinna's Functions and First Main Theorem
  10. Calculus Lemma
  11. Second Main Theorem
  12. Carlson-Griffiths Theory for Parabolic Complete Kähler Manifolds with Nonnegative Ricci Curvature
  13. Construction of $\Delta(r)$
  14. Nevanlinna's Functions and First Main Theorem
  15. Calculus Lemma
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $M$ be an $m$-dimensional non-parabolic complete noncompact Kähler manifold with nonnegative Ricci curvature. Let $(N, \omega)$ be a compact Kähler manifold of complex dimension $n\leq m.$ Let $D_1,\cdots, D_q$ be effective divisors in general position on $N$ such that each $D_j$ is cohomologous holds for all $r>0$ outside $E_\delta.$$\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (51)

  • Theorem 1.1: =Theorem \ref{['main']}
  • Corollary 1.2: Defect Relation
  • Corollary 1.3: Picard-type Theorem
  • Theorem 1.4: =Theorem \ref{['main220']}
  • Corollary 1.5: Defect Relation
  • Corollary 1.6: Picard-type Theorem
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 41 more