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Value distribution of meromorphic mappings on complete Kähler connected sums with non-parabolic ends

Xianjing Dong

TL;DR

The paper extends Nevanlinna theory to complete Kähler connected sums with non-parabolic ends, studying the value distribution of meromorphic maps $f:M\to X$ into projective manifolds. It constructs a heat-kernel–driven framework for Nevanlinna functions $T_f(r,L)$, $m_f(r,D)$, and $N_f(r,D)$ and proves a second main theorem with a defect term $\Xi(r,\delta,\kappa)$ under geometric conditions including PH and Ricci lower bounds. The authors derive defect relations and Picard-type conclusions, showing that Picard's little theorem holds for meromorphic maps under a volume-growth condition, and that Liouville rigidity for holomorphic functions follows from the Cauchy-Riemann equations in this setting. These results significantly generalize Nevanlinna theory to non-parabolic ends and clarify when Liouville rigidity and Picard-type phenomena persist on complicated global geometries. The work provides concrete classes of examples (Hermitian and Kähler connected sums) where the theory applies, highlighting the role of heat kernel and volume-growth in complex-analytic rigidity phenomena.

Abstract

All harmonic functions on $\mathbb C^m$ possess Liouville's property, which is well-known as the Liouville's theorem. In 1979, Kuz'menko and Molchanov discovered a phenomenon that the Liouville's property is not rigid for some harmonic functions on the connected sum $\mathbb C^m\#\mathbb C^m,$ where there exist a large number of non-constant bounded harmonic functions. This discovery motivates us to explore conditions under which harmonic functions possess Liouville's property. In this paper, we discuss the value distribution of meromorphic mappings from complete Kähler connected sums with non-parabolic ends into complex projective manifolds. Under a geometric condition, we establish a second main theorem in Nevanlinna theory. As a consequence, we prove that the Cauchy-Riemann equation ensures the rigidity of Liouville's property for harmonic functions if such connected sums satisfy a volume growth condition.

Value distribution of meromorphic mappings on complete Kähler connected sums with non-parabolic ends

TL;DR

The paper extends Nevanlinna theory to complete Kähler connected sums with non-parabolic ends, studying the value distribution of meromorphic maps into projective manifolds. It constructs a heat-kernel–driven framework for Nevanlinna functions , , and and proves a second main theorem with a defect term under geometric conditions including PH and Ricci lower bounds. The authors derive defect relations and Picard-type conclusions, showing that Picard's little theorem holds for meromorphic maps under a volume-growth condition, and that Liouville rigidity for holomorphic functions follows from the Cauchy-Riemann equations in this setting. These results significantly generalize Nevanlinna theory to non-parabolic ends and clarify when Liouville rigidity and Picard-type phenomena persist on complicated global geometries. The work provides concrete classes of examples (Hermitian and Kähler connected sums) where the theory applies, highlighting the role of heat kernel and volume-growth in complex-analytic rigidity phenomena.

Abstract

All harmonic functions on possess Liouville's property, which is well-known as the Liouville's theorem. In 1979, Kuz'menko and Molchanov discovered a phenomenon that the Liouville's property is not rigid for some harmonic functions on the connected sum where there exist a large number of non-constant bounded harmonic functions. This discovery motivates us to explore conditions under which harmonic functions possess Liouville's property. In this paper, we discuss the value distribution of meromorphic mappings from complete Kähler connected sums with non-parabolic ends into complex projective manifolds. Under a geometric condition, we establish a second main theorem in Nevanlinna theory. As a consequence, we prove that the Cauchy-Riemann equation ensures the rigidity of Liouville's property for harmonic functions if such connected sums satisfy a volume growth condition.
Paper Structure (26 sections, 34 theorems, 228 equations, 3 figures)

This paper contains 26 sections, 34 theorems, 228 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main Results
  4. Preliminaries
  5. Weighted Manifolds
  6. Faber-Krahn Inequality and Harnack Inequality
  7. Faber-Krahn Inequality
  8. Harnack Inequality
  9. Connected Sums of Riemannian Manifolds
  10. Almost Complex Structures of Connected Sums
  11. Heat Kernel on Connected Sums
  12. Setups on Complete Kähler Connected Sums
  13. Itô Formula
  14. Complete Kähler Connected Sums
  15. Nevanlinna's Functions via Heat Kernel
  16. ...and 11 more sections

Key Result

Theorem 1.1

Let $M=M_1\#\cdots\#M_\vartheta$ be a complete Kähler connected sum with $\vartheta$ non-parabolic ends, where all $M_j^{,}s$ are non-compact manifolds satisfying $(PH)$ in Property $ap2,$ with Ricci curvature bounded from below by a constant. Let $X$ be a complex projective manifold of complex dime holds for all $r>0$ outside $E_\delta,$ where $\Xi(r,\delta,\kappa)$ is defined by $(XI).$

Figures (3)

  • Figure 1: $\mathbb R^m\#\mathbb R^m$
  • Figure 2: Catenoid
  • Figure 3: Manifold with ends

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Corollary 1.7
  • Corollary 1.8
  • Corollary 1.9
  • Remark 1
  • ...and 43 more