Value Distribution Theory for Meromorphic Maps

Value distribution theory studies the behavior of mermorphic maps.

Author: Wilhelm Stoll

Publisher: Springer Science & Business Media

ISBN: 9783663052920

Category: Science

Page: 347

View: 760


Value distribution theory studies the behavior of mermorphic maps. Let f: M - N be a merom orphic map between complex manifolds. A target family CI ~ (Ea1aEA of analytic subsets Ea of N is given where A is a connected. compact complex manifold. The behavior of the inverse 1 family ["'(CI) = (f- {E )laEA is investigated. A substantial theory has been a created by many contributors. Usually the targets Ea stay fixed. However we can consider a finite set IJ of meromorphic maps g : M - A and study the incidence f{z) E Eg(z) for z E M and some g E IJ. Here we investigate this situation: M is a parabolic manifold of dimension m and N = lP n is the n-dimensional projective space. The family of hyperplanes in lP n is the target family parameterized by the dual projective space lP* We obtain a Nevanlinna theory consisting of several n First Main Theorems. Second Main Theorems and Defect Relations and extend recent work by B. Shiffman and by S. Mori. We use the Ahlfors-Weyl theory modified by the curvature method of Cowen and Griffiths. The Introduction consists of two parts. In Part A. we sketch the theory for fixed targets to provide background for those who are familar with complex analysis but are not acquainted with value distribution theory.

Value Distribution Theory of the Gauss Map of Minimal Surfaces in Rm

B. V. Shabat, Distribution of values of holomorphic mappings, Transl. Math. Monographs Vol. 61, AMS, 1985. W. Stoll, Introduction to value distribution theory of meromorphic maps, lecture notes in Math.

Author: Hirotaka Fujimoto

Publisher: Springer Science & Business Media

ISBN: 9783322802712

Category: Mathematics

Page: 208

View: 932


This book presents in a systematic and almost self-contained way the striking analogy between classical function theory, in particular the value distribution theory of holomorphic curves in projective space, on the one hand, and important and beautiful properties of the Gauss map of minimal surfaces on the other hand. Both theories are developed in the text, including many results of recent research. The relations and analogies between them become completely clear. The book is written for interested graduate students and mathematicians, who want to become more familiar with this modern development in the two classical areas of mathematics, but also for those, who intend to do further research on minimal surfaces.

Value Distribution Theory

THE AHLFORS - WEYL THEORY OF MEROMORPHIC MAPS ON PARABOLIC MANIFOLDS Wilhelm Stoll +) Contents. 1. Preface and review of value distribution in one variable 1 0 1 2. Hermitian geometry 1 O 6 3. Hermitian line bundles 113 4.

Author: I. Laine

Publisher: Springer

ISBN: 9783540394808

Category: Mathematics

Page: 250

View: 197


Value Distribution Theory

Stoll, W. [1] Introduction to value distribution theory of meromorphic maps, Lecture Notes in Math., no. 950, Springer, 1982, 210–359. [2] Value Distribution Theory for Meromorphic Maps. Vieweg, Braunschweig, 1985.

Author: Yang Lo

Publisher: Springer Science & Business Media

ISBN: 9783662029152

Category: Mathematics

Page: 269

View: 511


It is well known that solving certain theoretical or practical problems often depends on exploring the behavior of the roots of an equation such as (1) J(z) = a, where J(z) is an entire or meromorphic function and a is a complex value. It is especially important to investigate the number n(r, J = a) of the roots of (1) and their distribution in a disk Izl ~ r, each root being counted with its multiplicity. It was the research on such topics that raised the curtain on the theory of value distribution of entire or meromorphic functions. In the last century, the famous mathematician E. Picard obtained the pathbreaking result: Any non-constant entire function J(z) must take every finite complex value infinitely many times, with at most one excep tion. Later, E. Borel, by introducing the concept of the order of an entire function, gave the above result a more precise formulation as follows. An entire function J (z) of order A( 0 A

The Defect Relation of Meromorphic Maps on Parabolic Manifolds

[2] E. Bardis, The Defect Relation for Meromorphic Maps Defined on Covering Parabolic Manifolds, Thesis, Notre Dame (1990). [3]. ... [12] W. Stoll, Value Distribution Theory for Meromorphic Maps, Aspekte der Mathematik, E7 (1985).

Author: George Lawrence Ashline

Publisher: American Mathematical Soc.

ISBN: 9780821810699

Category: Mathematics

Page: 78

View: 927


This book is intended for graduate students and research mathematicians working in several complex variables and analytic spaces.

Contributions to Several Complex Variables

Wilhelm Stol] Value Distribution Theory for Meromorphic Maps 1985. X11, 347 pp. 16,2 X 22,9 cm. (Aspects of Mathematics, Vol. E7, ed by Klas Diederich.) Softcover Contents: Preface – Letters – Introduction – Value Distribution TheOry ...

Author: Alan Howard

Publisher: Springer Science & Business Media

ISBN: 9783663068167

Category: Mathematics

Page: 353

View: 312


In 1960 Wilhelm Stoll joined the University of Notre Dame faculty as Professor of Mathematics, and in October, 1984 the university acknowledged his many years of distinguished service by holding a conference in complex analysis in his honour. This volume is the proceedings of that conference. It was our priviledge to serve, along with Nancy K. Stanton, as conference organizers. We are grateful to the College of Science of the University of Notre Dame and to the National Science Foundation for their support. In the course of a career that has included the publication of over sixty research articles and the supervision of eighteen doctoral students, Wilhelm Stoll has won the affection and respect of his colleagues for his diligence, integrity and humaneness. The influence of his ideas and insights and the subsequent investigations they have inspired is attested to by several of the articles in the volume. On behalf of the conference partipants and contributors to this volume, we wish Wilhelm Stoll many more years of happy and devoted service to mathematics. Alan Howard Pit-Mann Wong VII III ~ c: ... ~ c: o U CI> .r. ~ .... o e ::J ~ o a:: a. ::J o ... (.!:J VIII '" Q) g> a. '" Q) E z '" ..... o Q) E Q) ..c eX IX Participants on the Group Picture Qi-keng LU, Professor, Chinese Academy of Science, Peking, China.

Value Distribution Theory and Related Topics

Japan 40 (1988), 237–249. Fujimoto H., Value distribution theory of the Gauss map of minimal surfaces in Friedr. Vieweg & Sohn, Braunschweig, 1993. Green M., Holomorphic maps into complex projective space omitting hyperplanes, Trans.

Author: Grigor A. Barsegian

Publisher: Springer Science & Business Media

ISBN: 9781402079511

Category: Mathematics

Page: 333

View: 326


The Nevanlinna theory of value distribution of meromorphic functions, one of the milestones of complex analysis during the last century, was c- ated to extend the classical results concerning the distribution of of entire functions to the more general setting of meromorphic functions. Later on, a similar reasoning has been applied to algebroid functions, subharmonic functions and meromorphic functions on Riemann surfaces as well as to - alytic functions of several complex variables, holomorphic and meromorphic mappings and to the theory of minimal surfaces. Moreover, several appli- tions of the theory have been exploited, including complex differential and functional equations, complex dynamics and Diophantine equations. The main emphasis of this collection is to direct attention to a number of recently developed novel ideas and generalizations that relate to the - velopment of value distribution theory and its applications. In particular, we mean a recent theory that replaces the conventional consideration of counting within a disc by an analysis of their geometric locations. Another such example is presented by the generalizations of the second main theorem to higher dimensional cases by using the jet theory. Moreover, s- ilar ideas apparently may be applied to several related areas as well, such as to partial differential equations and to differential geometry. Indeed, most of these applications go back to the problem of analyzing zeros of certain complex or real functions, meaning in fact to investigate level sets or level surfaces.

Nevanlinna Theory And Its Relation To Diophantine Approximation Second Edition

Defect relations for holomorphic maps between spaces of different dimensions, Duke Math. J.. 55, 213-251. Siu, Y.T. (1990). Nonequidimensional value distribution theory and meromorphic connections, Duke Math. J. 61, 341-367.

Author: Min Ru

Publisher: World Scientific

ISBN: 9789811233524

Category: Mathematics

Page: 444

View: 497


This book describes the theories and developments in Nevanlinna theory and Diophantine approximation. Although these two subjects belong to the different areas: one in complex analysis and one in number theory, it has been discovered that a number of striking similarities exist between these two subjects. A growing understanding of these connections has led to significant advances in both fields. Outstanding conjectures from decades ago are being solved.Over the past 20 years since the first edition appeared, there have been many new and significant developments. The new edition greatly expands the materials. In addition, three new chapters were added. In particular, the theory of algebraic curves, as well as the algebraic hyperbolicity, which provided the motivation for the Nevanlinna theory.

Nevanlinna s Theory of Value Distribution

... The Ahlfors–Weyl theory of meromorphic maps on parabolic manifolds, in I. Laine and S. Rickman, editors, Value distribution theory, ... S. TOPPILA, On the counting function for the a-values of a meromorphic function, Ann. Acad. Sci.

Author: William Cherry

Publisher: Springer Science & Business Media

ISBN: 9783662125908

Category: Mathematics

Page: 203

View: 255


This monograph serves as a self-contained introduction to Nevanlinna's theory of value distribution as well as a valuable reference for research specialists. Authors present, for the first time in book form, the most modern and refined versions of the Second Main Theorem with precise error terms, in both the geometric and logarithmic derivative based approaches. A unique feature of the monograph is its number theoretic digressions These special sections assume no background in number theory and explore the exciting interconnections between Nevanlinna theory and the theory of Diophantine approximation.

Unicity of Meromorphic Mappings

[242] Stoll, W., Introduction to value distribution theory of meromorphic maps, Lecture Notes in Math.950 (1982), 210-359, Springer-Verlag. [243] Stoll, W., The Ahlfors-Weyl theory of meromorphic maps on parabolic manifolds, ...

Author: Pei-Chu Hu

Publisher: Springer Science & Business Media

ISBN: 9781475737752

Category: Mathematics

Page: 467

View: 583


For a given meromorphic function I(z) and an arbitrary value a, Nevanlinna's value distribution theory, which can be derived from the well known Poisson-Jensen for mula, deals with relationships between the growth of the function and quantitative estimations of the roots of the equation: 1 (z) - a = O. In the 1920s as an application of the celebrated Nevanlinna's value distribution theory of meromorphic functions, R. Nevanlinna [188] himself proved that for two nonconstant meromorphic func tions I, 9 and five distinctive values ai (i = 1,2,3,4,5) in the extended plane, if 1 1- (ai) = g-l(ai) 1M (ignoring multiplicities) for i = 1,2,3,4,5, then 1 = g. Fur 1 thermore, if 1- (ai) = g-l(ai) CM (counting multiplicities) for i = 1,2,3 and 4, then 1 = L(g), where L denotes a suitable Mobius transformation. Then in the 19708, F. Gross and C. C. Yang started to study the similar but more general questions of two functions that share sets of values. For instance, they proved that if 1 and 9 are two nonconstant entire functions and 8 , 82 and 83 are three distinctive finite sets such 1 1 that 1- (8 ) = g-1(8 ) CM for i = 1,2,3, then 1 = g.