The Uniqueness Problem Of Meromorphic Mappings In Several Complex Variables

The value distribution theory,the great branch of analysis,was initiated by R.Nevanlinna[1].It has been a great theory in the analysis,and the results of the theory of value distribution permeate almost all aspects of mathemat-ics.Starting with Nevanlinna's paper,R.Nevanlinna,Cartan,Bloch,Weyl,Ahlfors,Stoll,Vojta,Fujimoto and others began to carry out the study of value distribution.After that,the value distribution theory developed a lot.Many of the profound and beautiful achievements have emerged.Besides,many results in the value distribution theory have become a powerful research tool for many branches.This phenomenon has enriched their research methods and greatly promoted their development.In many areas such as uniqueness theory,differential equations,diophantine approximation,complex dynamical systems,the number theory,criteria for normal family,hyperbolic geometry,many important developments are inseparable from the theory of value dis-tribution.The uniqueness problem occupies an important position in the value dis-tribution theory.In this area,we study how to decide a the only meromorphic function or mapping by the conditions of the distribution of values.Nevanlin-na[2]firstly study the uniqueness problems of meromorphic function by using the value distribution theory,and he gained Nevanlinna five-value theorem of the single complex meromorphic function.The uniqueness of meromor-phic mappings from Cm to Pn(C)sharing hyperplanes began in 1975,when Fujimoto,a Japanese mathematician,proved several important results.Those results can be seen as the five-value theorem of high-dimensional case.In the subsequent development of the subject,Smiley,Dethloff,Risto Korhonen and others have also made some results.On the basis of them,we gain two uniqueness theorems of meromorphic mappings by using their method.The specific arrangement is as follows:The first chapter introduces the basic knowledge of high dimensional value distribution and the basic concept of sharing hyperplane problem,Including:related concepts on hyperplane,basic symbols in value distribution,count-ing functions in value distribution,the characteristic function,the first main theorem and the second main theorem about sharing hyperplanes and so on.In the second chapter,we obtain two uniqueness results of sharing hyper-planes.By changing the total sharing condition to partial sharing condition,we improved the result of Tingbin Cao and Hongxun Yi.On the other hand,by means of changing the partial sharing condition to total sharing condition,and removing a limit condition,we improve a uniqueness result of Feng Lv.The details are as follows:Theorem 2.3 Let f,g be two Lmap from Cm to Pn(C),and let Hj(1 ?j?q)be q Gplane,when i ? j,dim f-1(Hi ? Hj)? m-2.Let mj(j =1,2,...,q)be q positive integer or ?,s.t.m1>m2...?mq?n.Supposing that and when z ? Uj=1q{z ?Cm| 0<v(g,Hj)? mj},f(z)=q(z),Supposing that then f(z)=g(z).Theorem 2.6 Let f,g be two Lmap from Cm to Pn(C),and let Hj(1?j?q)be q Gplane,when i ? j,dimf-1(Hi ? Hj)? m-2.Let mj(j =1,2,…,q)be q positive integer or ?,s.t.m1,m2...,mq?n,Supposing that and when z?Uj=1{z ?Cm|0<v(f,Hj)?mj},f(z)=g(z),If q>2n + 2 +(?),then f=gIn chapter 3,we present our outlook on uniqueness problems.