arXiv subjects

Xianjing Dong

Publications and source records attributed to Xianjing Dong.

At least 19 recordsLinked to original sources

arXiv2026

On Picard's Problem via Nevanlinna Theory II

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.

math.CV↗

arXiv2026

On Picard's Problem via Nevanlinna Theory

We consider the classical Picard's problem for non-parabolic complete Kähler 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.

math.CV↗

arXiv2025

Algebroid Mappings and Their Equidistribution Theory

In this paper, the concept of algebroid mappings of complex manifolds is introduced based on that a large number of complex systems of PDEs admit multi-valued solutions that can be defined by a system of independent algebraic equations over the field of meromorphic functions. It is well-known that Nevanlinna theory is an important tool in complex ODE theory. To develop a similar tool applied to the study of algebroid solutions of complex systems of PDEs, one explores the equidistribution theory of algebroid mappings. Via uniformizating an algebroid mapping, we obtain a second main theorem of algebroid mappings from a complete Kähler manifold into a complex projective manifold provided that some certain conditions are imposed.

math.CV↗

arXiv2025

A Generalization of Zalcman's Lemma on Complex Lie Groups

Zalcman's Lemma makes significant applications in normal families, complex dynamics and related problems in complex analysis. In the present paper, we are devoted to generalizing the classical Zalcman's lemma to complex Lie groups by means of exponential mappings defined by holomorphic one-parameter subgroups.

math.CV↗

arXiv2025

Nevanlinna Theory of Algebroid Functions on Complete Kähler Manifolds

In this paper, we generalize the classical Nevanlinna theory of algebroid functions from $\mathbb C$ to a complete Kähler manifold with either non-negative Ricci curvature or non-positive sectional curvature. As its applications, we establish some Picard type theorems and five-value type theorems for algebroid functions under certain conditions.

math.CV↗

arXiv2025

A refined form of the second main theorem on complete non-positively curved Kähler manifolds

How to devise a second main theorem with best error terms is a central problem in the study of Nevanlinna theory. However, it seems difficult to be done for a general non-positively curved Kähler manifold. Based on the work of A. Atsuji in Nevanlinna theory, we present a refined form of the second main theorem of meromorphic mappings on a general complete Kähler manifold with non-positive sectional curvature using a good estimate. This result improves the error terms in the second main theorem obtained by A. Atsuji in 2018.

math.CV↗

arXiv2025

Nevanlinna theory on complete Kähler manifolds with non-negative Ricci curvature

The paper develops an equidistribution theory of meromorphic mappings from a complete Kähler manifold with non-negative Ricci curvature into a complex projective manifold intersecting normal crossing divisors. When the domain manifolds are of maximal volume growth, one obtains a second main theorem with a refined error term. As a result, we prove a sharp defect relation in Nevanlinna theory. Furthermore, our results are applied to the propagation problems of algebraic dependence. As major consequences, we set up several unicity theorems for dominant meromorphic mappings on complete Kähler manifolds. In particular, we prove a five-value theorem on complete Kähler manifolds, which gives an extension of Nevanlinna's five-value theorem for meromorphic functions on $\mathbb C.$

math.CV↗

arXiv2025

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

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.

math.CV↗

arXiv2024

Nevanlinna Theory on Geodesic Balls of Complete Kähler Manifolds

We study Nevanlinna theory of meromorphic mappings from a geodesic ball of a general complete Kähler manifold with non-negative Ricci curvature into a complex projective manifold by introducing a heat kernel method. When dimension of a target manifold is not greater than one of a source manifold, we establish a second main theorem which is a generalization of the classical second main theorem for a ball of $\mathbb C^m.$ If a source manifold is non-compact and it carries a positive global Green function, then we establish a global second main theorem for the source manifold. As a result, we obtain a Picard's theorem for complete Kähler manifolds with non-negative Ricci curvature.

math.CV↗

arXiv2024

Unicity problem on meromorphic mappings of complete Kahler manifolds

Nevanlinna's unicity theorems have always held an important position in value distribution theory. The main purpose of this paper is to generalize the classical Nevanlinna's unicity theorems to non-compact complete Kahler manifolds with nonpositive sectional curvature or nonnegative Ricci curvature.

math.DG↗

arXiv2023

Nevanlinna's five-value theorem on non-positively curved complete Kähler manifolds

Nevanlinna's five-value theorem is well-known as a famous theorem in value distribution theory, which asserts that two non-constant meromorphic functions on $\mathbb C$ are identical if they share five distinct values ignoring multiplicities in $\overline{\mathbb C}.$ The central goal of this paper is to generalize Nevanlinna's five-value theorem to non-compact complete Kähler manifolds with non-positive sectional curvature by means of the theory of algebraic dependence. With a certain growth condition imposed, we show that two nonconstant meromorphic functions on such class of manifolds are identical if they share five distinct values ignoring multiplicities in $\overline{\mathbb C}.$

math.CV↗

arXiv2022

Nevanlinna-type theory based on heat diffusion

We obtain an analogue of Nevanlinna theory of holomorphic mappings from a complete and stochastically complete Kähler manifold into a complex projective manifold. When certain curvature conditions are imposed, the Nevanlinna-type defect relation based on heat diffusion is derived.

math.CV↗

arXiv2022

Nevanlinna theory via holomorphic forms

This paper re-develops the Nevanlinna theory for meromorphic functions on $\mathbb C$ in the viewpoint of holomorphic forms. According to our observation, Nevanlinna's functions can be formulated by a holomorphic form. Applying this thought to Riemann surfaces, one then extends the definition of Nevanlinna's functions using a holomorphic form $\mathscr S$. With the new settings, an analogue of Nevanlinna theory on \emph{weak $\mathscr S$-exhausted Riemann surfaces} is obtained, which is viewed as a generalization of the classical Nevanlinna theory on $\mathbb C$ and $\mathbb D.$

math.CV↗

arXiv2022

The Second Main Theorem for spherically symmetric Kähler manifolds

We investigate the value distribution of holomorphic maps defined on one class of Kähler manifolds. With the very natural settings, we establish a Second Main Theorem which is of the similar form as ones of the classical Second Main Theorem for complex Euclidean spaces and complex unit balls.

math.CV↗

arXiv2022

On Griffiths conjecture

By using techniques of holomorphic jets and Jacobian fields, we devise a non-equidistribution theory of holomorphic curves into complex projective varieties intersecting normal crossing divisors. Based on this theory established, we prove the Griffiths conjecture and the Green-Griffiths conjecture in Nevanlinna theory and algebraic geometry.

math.CV↗

arXiv2021

Algebraic degeneracy of holomorphic curves

We consider the algebraic degeneracy of holomorphic curves from a point of view of meromorphic vector fields. Employing the notion of Jocabian sections introduced by W. Stoll, we establish a Second Main Theorem type inequality. As consequences, several algebraic degeneracy theorems are obtained for holomorphic curves into a complex projective variety.

math.CV↗

arXiv2021

Holomorphic curves in moduli spaces of polarized Abelian varieties

We study the value distribution of holomorphic curves from a general open Riemann surface into a smooth logarithmic pair $(X, D).$ By stochastic calculus, we first obtain a version of tautological inequality (proposed by McQuillan) and a logarithmic derivative lemma. Then, one uses them to establish a Second Main Theorem of Nevanlinna theory for pair $(X, D)$ under certain conditions. Finally, we apply the Second Main Theorem to study the holomorphic curves from a general open Riemann surface into certain special moduli spaces of polarized Abelian varieties intersecting boundary divisors.

math.CV↗

arXiv2021

Nevanlinna theory on complete Kähler manifolds

We study Nevanlinna theory on complete Kähler manifolds. As a consequence of the main result, we prove a defect relation of holomorphic mappings from complete Kähler manifolds of non-positive sectional curvature into complex projective manifolds under certain growth condition.

math.CV↗