Value distribution theory studies the behavior of mermorphic maps.

Author: Wilhelm Stoll

Publisher: Springer Science & Business Media

ISBN: 9783663052920

Category: Science

Page: 347

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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.

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

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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.

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

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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

Shiffman, B. : Holomorphic and meromorphic mappings and curvature. Math. Ann. 222 (1976) , l 71 – 19 4. Shiffman, B. : Applications of geometric measure theory to value distribution theory for meromorphic maps, in "Value-Distribution ...

Author: I. Laine

Publisher: Springer

ISBN: 9783540394808

Category: Mathematics

Page: 250

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[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

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

Soc. Japan 34 (1982), 527–539. Fujimoto H., Finiteness of some families of meromorphic maps, Kodai Math. J. 11 (1988), 47–63. Fujimoto H., Uniqueness problem with truncated multiplicities in value distribution theory I, II, Nagoya Math.

Author: Grigor A. Barsegian

Publisher: Springer Science & Business Media

ISBN: 9781402079511

Category: Mathematics

Page: 333

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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.

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

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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.

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

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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.

... [73] [711] [75] [76] [77] Shiffman, B. : Applications of geometric measure theory to value distribution theory for meromorphic maps. Value-Distribution Theory Part A (Edited by R. O. Kujala and A. L. Witter III) Pure and Appl. Math.

Author: W. Stoll

Publisher: Springer

ISBN: 9783540373063

Category: Mathematics

Page: 216

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SHIFFMAN , BERNARD [ 1 ] Extension of positive line bundles and meromorphic maps , Invent . Math . 15 ( 1972 ) , 332-347 . [ 2 ] Applications of geometric measure theory to value distribution theory for meromorphic maps , Value ...

Author: Boris Vladimirovich Shabat

Publisher: American Mathematical Soc.

ISBN: 0821898116

Category: Mathematics

Page: 236

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A vast literature has grown up around the value distribution theory of meromorphic functions, synthesized by Rolf Nevanlinna in the 1920s and singled out by Hermann Weyl as one of the greatest mathematical achievements of this century. The multidimensional aspect, involving the distribution of inverse images of analytic sets under holomorphic mappings of complex manifolds, has not been fully treated in the literature. This volume thus provides a valuable introduction to multivariate value distribution theory and a survey of some of its results, rich in relations to both algebraic and differential geometry and surely one of the most important branches of the modern geometric theory of functions of a complex variable. Since the book begins with preparatory material from the contemporary geometric theory of functions, only a familiarity with the elements of multidimensional complex analysis is necessary background to understand the topic. After proving the two main theorems of value distribution theory, the author goes on to investigate further the theory of holomorphic curves and to provide generalizations and applications of the main theorems, focusing chiefly on the work of Soviet mathematicians.