Nevanlinna Theory And Two Classes Of Complex Linear Differential-difference Equations

Nevanlinna theory(cf.[35,43,46])is generalization of Picard's(little)theorem and part of theory of meromorphic functions,which was devised by R.Nevanlinna in 1925.A fundamental tool to study this theory is the Nevanlinna's characteristic Tf(r)which measures the rate of the growth of a meromorphic function f.This theory was later generalized into meromorphic functions on Cn and parabolic Riemann surfaces,and holomorphic curves from C into Pm(C)and into algebraic varieties(cf.[43,46]).Efforts were also made in order to extend the theory into the higher dimensional manifolds,for example,some important work was done by W.Stoll(cf.[51,53]).Recently,A.Atsuji(cf.[2,3]),based on the Brownian motion,investigated the value distribution of meromorphic functions whose domain manifolds are complete Kahler manifolds.For example,Atsuji(cf.[3])in 2008 established the second main theorem for meromorphic functions on submanifolds in Cm,and later established the second main theorem(cf.[2])on complete Kahler manifolds by using the heat diffusion method(the probabilistic technique).For instance,he provedTheorem 0.0-4(Atsuji[3])Let f be a nonconstant meromorphic function on com- plete Kahler manifold M and let a1,…,aq be distinct points in P1(C).Then,for any?>0,we have (?)mf(r,aj)+ N1(r,f)?2T(r)+2Nf(r,Ric)+O(logTf(r))+log C(o,r,?) holds except for r in an exceptional set E? of finite length.However,the term C(o,r,?)appeared needs some estimates for Green's functions,which is very hard to compute.Furthermore,in general,it could be very large.To get rid of this term,Atsuji[7]in 2010 introduced the new notions of mx(t,a)Nx(t,a)and Tx(t)(cf.[7]for the definitions),which are the analogy of the classical Nevanlinna's functions.With these notations,Atsuji[7]proved an analogy of Nevnalinna's Second Main Theorem.To state this theorem,we recall the concept of that M is stochastically complete.Let M be a(geodesically)complete Riemann manifold with Laplace-Beltrami operator ? with respect to its metric,and let p(t,x,y)be the minimal fundamental solution of the heat equation(?)u/(?)t-1/2?u=0.M is said to be stochastically complete if ?MP(t,x,y)dV(y)=1 holds for all x ? M,where dV is the Riemannian volume measure of M.Theorem 0.0.5(Atsuji[7])Let f be a nonconstant meromorphic function on com-plete and stochastically complete Kahler manifold M,and let …,aq be distinct points in P1(C).Assume that Tx(t)<? for 0<t<oo and limt?? Tx(t)=?,as well as |Nx(t,Ric)|<?.Then,for any ?>0,we have (?)mx(t,aj)+N1(t,x)?2Tx(t)+2Nx(t,Ric)+(4q+1+?)logTx(t)+O(1) holds except for t in an exceptional set E?(?)[0,?)of finite length.Remark 0.0.6 In the original conclusion(Theorem 17,[7]),the coefficient of log Tx(t) in the above theorem is 5+? not 4q+1+?,but we think it should be 4q+1+? in our verification.In this thesis,we introduce the Nevanlinna theory by using stochastic calculus,fol-lowing the works by B.Davis,T.K.Carne and A.Atsuji and etc..We provide(another) proofs for the classical results of meromorphic functions and results by Cartan-Ahlfors of holomorphic curves by applying the probabilistic method,we also extend Atsuji s results into a product of complete Kahler manifolds.Now let's turn to the topics of complex linear differential-difference equations.It is a fundamental problem to study the structure of solutions to the linear differential-difference equation with constant coefficients,written as follows(?)Aklf(k)(z-?l)=g(z),where Akl,?l are the complex constants.The problem for structure of this equation has been focused on by many authors[10,15,18,44,57].However,such problem has not yet been finally settled.For example,we do not know whether there exists an entire solution of order exceeding 1,as well as the structure of general entire solutions.It was proposed by C.C.Yang in 1970s as followsProblem(Yang)Is there an entire solution of order exceeds one to the following linear complex differential-difference equation f'(z)=f(z+1)and what are the general entire solutions?For the problem,we give the structure of entire solutions of the equation in Chapter 4.Note that a linear differential-difference equation can be regarded as an infinite-order linear differential equation.In Chapter 4,we investigate the structure of entire solutions,but we do not investigate the existence and uniqueness of entire solutions.So,we have an investigation to the the existence and uniqueness of entire solutions of a class of infinite-order linear differential-difference equations in Chapter 5.Note that such class can be viewed as the generalization of finite-order linear differential-difference equations.This thesis consists of five chapters in which the general ideas are summarized as follows:In Chapter 1,we introduce some basics including the classical Nevanalinna theory and stochastic theory and potential theory and entire function theory.In Chapter 2,we provide a probabilistic method(via the Brownian motion)in proving the classical Nevanlinna's first and second main theorems for both meromorphic functions and holomorphic curves.In Chapter 3,we establish the first and second main theorem for holomorphic maps from a product of complete Kahler manifolds into P1(C)based on the heat diffusion method developed by Atsuji[7].Some Ricci curvature conditions for domain manifolds are imposed in order to get meaningful results.With some conditions,we show a nonconstant meromorphic function can omit at most two points.In Chapter 4,we give the structure of the entire solutions of linear differential-difference equation with constant coefficients,written as(?)Aklf(k)(z+?l)=g(z)?As an important consequence,we answer the question proposed by C.C.Yang who asked that what are the general entire solutions of the equation f(z)=f(z+1)?Furthermore,our method extends an important result concerning the entire solutions of a class linear differential equations of infinite order obtained by G.Valiron.In Chapter 5,we investigate the existence and uniqueness problem for the entire solutions to a class of linear differential-difference equations of infinite order in terms of(*?)akf(vk)(z+?k)=h(z),where 0=v0?v1?…,a0,a1,…are complex constants and h(z)is an entire function of growth of exponential type.