# Value distribution and potential theory

@article{Eremenko2003ValueDA, title={Value distribution and potential theory}, author={Alexandre Eremenko}, journal={arXiv: Complex Variables}, year={2003} }

We describe some results of value distribution theory of holomorphic curves and quasiregular maps, which are obtained using potential theory. Among the results discussed are: extensions of Picard's theorems to quasiregular maps between Riemannian manifolds, a version of the Second Main Theorem of Nevanlinna for curves in projective space and non-linear divisors, description of extremal functions in Nevanlinna theory and results related to Cartan's 1928 conjecture on holomorphic curves in the…

## 17 Citations

### Value distribution of holomoprhic curves on an angular domain

- Mathematics
- 2015

In this paper, we investigate the value distribution of holomorphic curves on an angular domain from the point of view of potential theory and establish the first and second fundamental theorems…

### ON THE EXISTENCE OF SINGULAR DIRECTIONS OF HOLOMORPHIC MAPS FROM THE UNIT DISK INTO Pn（C）

- Mathematics
- 2010

### Landau's theorem for holomorphic curves in projective space and the Kobayashi metric on hyperplane complements

- Mathematics
- 2006

We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1 hyperplanes in general position in n-dimensional complex projective space, we find an explicit constant K such that for…

### LANDAU ’ S THEOREM FOR HOLOMORPHIC CURVES

- Mathematics
- 2008

We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1 hyperplanes in general position in Pn, we find an explicit constant K such that for every holomorphic map f from the unit…

### LANDAU’S THEOREM FOR HOLOMORPHIC CURVES IN PROJECTIVE SPACE AND THE KOBAYASHI METRIC ON HYPERPLANE COMPLEMENTS

- Mathematics
- 2007

We prove an effective version of a theorem of Dufresnoy: For any set of 2n+1 hyperplanes in general position in Pn, we find an explicit constant K such that for every holomorphic map f from the unit…

### An extension of Lewis's Lemma, renormalization of harmonic and analytic functions, and normal families

- Mathematics, Chemistry
- 2009

An extension of a lemma due to J. Lewis is established and is used to give rapid proofs of some classical theorems in complex function theory such as Montel's theorem and Miranda's theorem. Another…

### Value Distribution of Meromorphic Functions

- Mathematics
- 2008

Characteristics of the behavior of a meromorphic function and the first fundamental theorem Meromorphic functions of finite order The second fundamental theorem Deficient values Asymptotic properties…

### Normal criterion for families of holomorphic maps of several complex variables into PN (C) with moving hypersurfaces

- Mathematics
- 2009

### Proof of a conjecture of Pólya on the zeros of successive derivatives of real entire functions

- Mathematics
- 2006

We prove Pólya’s conjecture of 1943: For a real entire function of order greater than 2 with finitely many non-real zeros, the number of non-real zeros of the nth derivative tends to infinity, as…

### Value distribution of meromorphic functions

- Philosophy
- 2011

Preliminaries of Real Functions.- Characteristics of a Meromorphic Function.- T Directions of a Meromorphic Function.- Argument Distribution and Deficient Values.- Meromorphic Functions With Radially…

## References

SHOWING 1-10 OF 54 REFERENCES

### Nevanlinna Theory and Its Relation to Diophantine Approximation

- Mathematics
- 2001

Nevanlinna Theory for Meromorphic Functions and Roth's Theorem Holomorphic Curves into Compact Riemann Surfaces and Theorems of Siegel, Roth, and Faltings Holomorphic Curves in Pn(C) and Schmidt's…

### Extremal holomorphic curves for defect relations

- Mathematics
- 1998

Drasin’s theorem describing meromorphic functions of finite order with maximal sum of deficiencies is extended to holomorphic curves in projective space. A conjecture about holomorphic curves…

### The analogue of Picard's theorem for quasiregular mappings in dimension three

- Mathematics
- 1985

The theory of quasiregular mappings has turned out to be the fight extension of the geometric parts of the theory of analytic functions in the plane to real n-dimensional space. The study of these…

### PROOF OF A CONDITIONAL THEOREM OF LITTLEWOOD ON THE DISTRIBUTION OF VALUES OF ENTIRE FUNCTIONS

- Mathematics
- 1988

It is proved that for any entire function / of finite nonzero order there is a set S in the plane with density zero and such that for any a € C almost all the roots of the equation /(ζ) = α belong to…

### A Picard Type Theorem for Holomorphic Curves

- Mathematics
- 1999

Let P be complex projective space of dimension m, π : Cm+1\{0} → P the standard projection and M ⊂ P a closed subset (with respect to the usual topology of a real manifold of dimension 2m). A…

### Introduction to Complex Hyperbolic Spaces

- Mathematics
- 1987

0 Preliminaries.- I Basic Properties.- II Hyperbolic Imbeddings.- III Brody's Theorem.- IV Negative Curvature on Line Bundles.- V Curvature on Vector Bundles.- VI Nevanlinna Theory.- VII Applications…

### Analysis and Geometry on Groups

- Mathematics
- 1993

Preface Foreword 1. Introduction 2. Dimensional inequalities for semigroups of operators on the Lp spaces 3. Systems of vector fields satisfying Hormander's condition 4. The heat kernel on nilpotent…

### theory of differential forms on manifolds

- Mathematics
- 1995

In this paper, we establish a Hodge-type decomposition for the LP space of differential forms on closed (i.e., compact, oriented, smooth) Rieman- nian manifolds. Critical to the proof of this result…

### Metric Structures for Riemannian and Non-Riemannian Spaces

- Mathematics
- 1999

Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.-…

### On the riesz charge of the lower envelope of δ-subharmonic functions

- Mathematics
- 1992

By potential theoretic methods involving the Cartan fine topology a recent result by two of the authors is extended as follows: The Riesz charge of the lower envelope of a family of 3 or more…