Narrow Results

We prove a second main theorem type estimate in Nevanlinna theory when a target space is a family of curves. This estimate unifies the truncated q-small function theorem, and the height inequality for curves over function fields.

In 1983, Nochka proved a conjecture of Cartan on defects of holomorphic curves in ℂPⁿ relative to a possibly degenerate set of hyperplanes. In this paper, we generalize Nochka's theorem to the case of curves in a complex projective variety intersecting hypersurfaces in subgeneral position. Further work will be needed to determine the optimal notion of subgeneral position under which this result can hold, and to lower the effective truncation level which we achieved.

We prove a second main theorem type estimate in Nevanlinna theory for holomorphic curves f : Y → X from finite covering spaces Y → ℂ of the complex plane ℂ into complex projective manifolds X of maximal albanese dimension. If X is moreover of general type, then this implies that the special set of X is a proper subset of X. For a projective curve C in such X, our estimate also yields an upper bound of the ratio of the degree of C to the geometric genus of C, provided that C is not contained in a proper exceptional subset in X.

We construct families of hyperbolic hypersurfaces $X_d\subset {\mathbb{P}}^{n+1}(ℂ)$ of degree $d\ge {(\frac{n+3}{2})}^2$.

This paper derives several new results, as well as gives a partial survey of the recent development, on the value sharing for algebraic and holomorphic curves into ${\mathbb{P}}^n(ℂ)$.

In this paper, by introducing a new method in estimating the counting function of the auxiliary function, we prove a new generalization of uniqueness theorems for meromorphic mappings into ${\mathbb{P}}^n(ℂ)$ which share few hyperplanes regardless of multiplicities. Our result improves the previous results in this topic. Moreover, our proof is simpler than the previous proofs.

In this paper, under the refinement of the subgeneral position, we give an improvement for the Second Main Theorem with truncated counting functions of algebraically non-degenerate holomorphic curves into algebraic varieties $V$ intersecting divisors in subgeneral position with some index.

In this paper, we will give suitable conditions on differential polynomials $Q(f)$ such that they take every finite nonzero value infinitely often, where $f$ is a meromorphic function in complex plane. These results are related to Problems 1.19 and 1.20 in a book of Hayman and Lingham [Research Problems in Function Theory, preprint (2018), https://arxiv.org/pdf/1809.07200.pdf]. As consequences, we give a new proof of the Hayman conjecture. Moreover, our results allow differential polynomials $Q(f)$ to have some terms of any degree of $f$ and also the hypothesis $n>k$ in [Theorem 2 of W. Bergweiler and A. Eremenko, On the singularities of the inverse to a meromorphic function of finite order, Rev. Mat. Iberoamericana11(2) (1995) 355–373] is replaced by $n\ge 2$ in our result.

Motivated by the notion of the algebraic hyperbolicity, we introduce the notion of Nevanlinna hyperbolicity for a pair $(X,D)$, where $X$ is a projective variety and $D$ is an effective Cartier divisor on $X$. This notion links and unifies the Nevanlinna theory, the complex hyperbolicity (Brody and Kobayashi hyperbolicity), the big Picard-type extension theorem (more generally the Borel hyperbolicity). It also implies the algebraic hyperbolicity. The key is to use the Nevanlinna theory on parabolic Riemann surfaces recently developed by P$\mathrm{\text{ă}}$un and Sibony [Value distribution theory for parabolic Riemann surfaces, preprint (2014), arXiv:1403.6596].

In this paper, we incorporate the beta constants into the defect relation and deduce a non-integrated defect relation for meromorphic maps from a Kähler manifold whose universal covering is a ball into a projective variety intersecting general divisors properly.

Our goal in this paper is to establish some second main theorems for holomorphic mappings from angular domains into projective varieties intersecting arbitrary families of moving hypersurfaces with truncated counting functions. Our results generalize and also improve all previous results for the case of holomorphic mappings from the complex plane.

In this paper, we show some Second Main Theorems for zero-order meromorphic mappings intersecting slowly moving targets in ${\mathbb{P}}^m(ℂ)$ by considering their $p$-Casorati determinant. Our results are $p$-difference analogues of Cartan’s Second Main Theorem for moving targets. As an application, we give an unicity theorem for meromorphic mappings of ${ℂ}^m$ into ${\mathbb{P}}^m(ℂ)$ under the growth condition “order $=0$”.

In this paper, we will prove a uniqueness theorem in the case of meromorphic functions on the annuli share $q(q\ge 5)$ distinct elements with different multiple values.

This paper presents some new identities and inequalities in integral and differential geometry, in real and complex analysis in ODE. Ideas, methods and problems come from some preceding studies related to Gamma-lines. There are also certain bridges between the presented results and Nevanlinna theory of meromorphic functions.

Some purely geometric results analogous to the second fundamental theorems in the classical Nevanlinna and Ahlfors theories are revealed. These analogs are valid for arbitrary analytic (meromorphic) functions in given domains in contrast to the Nevanlinna and Ahlfors theories that are applicable only for some known subclasses of functions.