Meromorphic Solutions Of Fermat Type Functional Equation And The Gauss Map Of Complete Minimal Surfaces

This dissertation consists of three parts.The first part studies some issues concerning the existence of non-trivial meromorphic solutions for the Fermat type functional equationF8(z)+G8(z)+H8(z)=1,and the existence of non-trivial entire solutions for the Fermat type functional equationF6(z)+G6(z)+H6(z)=1,The following results have been obtained:·There does not exist non-trivial meromorphic solutions with the exponent of conver-gence of poles is less than one satisfying the equation F8(z)+G8(z)+H8(z)=1.·There does not exist non-trivial entire solutions with the exponent of convergence of zeros is less than one satisfying the equation F6(z)+G6(z)+H6(z)=1.The second part investigates the relation between meromorphic functions and the Gauss map of complete minimal surfaces,and several classes of meromorphic functions on C were obtained which can be viewed as the Gauss map of complete minimal surfaces.This partially answers the question proposed by F.Xavier and X.L.Chao:What conditions will guarantee that the meromorphic functions in the complex plane are the Gauss map of complete minimal surfaces.The following conditions have been drawn:·Meromorphic function in the complex plane with the property that either the exponent of convergence of zeros or the exponent of convergence of poles is less than 1/2,then it must be the Gauss map of some complete minimal surface.·If g1(z)and g2(z)≠0 are entire functions which have no common zeros,at least one of the primitve functions of g12(z)and g22(z)is a composite function,which is composed of a finite numbers of entire functions with order less than 1/2,then the meromorphic function g1(z)/g2(z)can be viewed as the Gauss map of some complete minimal surface.·If g1(z)and g2(z)≠0 are entire functions which have no common zeros at least one of their Taylor expansions at the origin has 2-order Fejer gaps,then the meromorphic function g1(z)/g2(z)can be viewed as the Gauss map of some complete minimal surface.The third part discusses the uniqueness of exponential polynomials.For the special class of entitre functions,the Nevanlinna’s five-value and four-value theorem have been improved in angular domain.The following results have been proved:·Suppose f(z),g(z)are non-constant exponential polynomials and ak(k=1,2,3,4)are distinct finite complex numbers.If K≥0 and angular domains Ωk(k=1,2,3,4)of opening is larger than π,for each k ∈ {1,2},f(z)and g(z)share ak CM in the domain Dk=Ωk∩{z∈C||z|>K},and for each j ∈ {3,4},f(z)and g(z)share aj IM in the domain Dj=Ωj∩{z∈C||z|>K},then f(z)≡g(z).·Suppose f(z),g(z)are non-constant exponential polynomials and ak(k=1,2,3,4)are distinct finite complex numbers.If K>0 and angular domains Ωk(k=1,2,3)of opening is larger than π,for each k∈{1,2},f(z)and g(z)share ak CM in the domain Dk=Ωk∩{z∈C||z|>K},and f(z)and g(z)share a3 IM in the domain D3=Ω3∩{z ∈ C ||z|>K},then there exists a linear function h(z),such that h(f(z))· h(g(z))=1.