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The Hartogs domain Bergman kernel function and Lu Qikeng problem

Author: LiuYuLan
Tutor: WangAn
School: Capital Normal University
Course: Basic mathematics
Keywords: Packet circular domain Bergman kernel function Lu Qikeng problem Lu Qikeng domain
CLC: O174.56
Type: Master's thesis
Year: 2008
Downloads: 14
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Abstract


In 1921 , Bergman S. The introduction of the concept of the Bergman kernel function in 1933 to promote the theory of the Bergman kernel function of several complex variables . Bergman kernel function theory provides a useful tool for a lot of research in the field of mathematics . For example , complex analysis , differential geometry , mathematical physics , and so on . Bounded domain of C n , how to find the explicit expression of the Bergman kernel function is not easy , and it has become an important direction in the study of several complex variables . Lu Qikeng problem stems from Lu Qikeng Bergman kernel function proposed in a 1966 article , whether zero problems . This problem since the introduction of the so far aroused the interest of many mathematicians , and gives a lot of examples , but the examples given are counter-examples . Bergman kernel function of the domain , you can know that zero distribution , so as to judge whether the two domains holomorphic equivalence provides a powerful tool by the research domain Lu Qikeng problem . The main results of this paper : (1) we obtain the domain Bergman kernel function in this section, we first the domain Ω (1 N 1 N 2 < / sub> ; K , L ) is holomorphic self isomorphic transformation and complete the standard orthogonal function system requirements Bergman kernel function , followed by the expansion principle to calculate the domain Ω ( N 0 < / sub > N 1 < / sub> N 2 ; K, L) Bergman kernel function . This approach not only simplifies the calculation of high dimensional complex space bounded domain Bergman kernel function was an expression of the process , but can be extended in the end space is any finite direct product of the form contains the origin bounded domain transitive Hartogs domain . ( 2 ) to discuss the domain of Lu Qikeng problems . Here, we use of the domain Ω holomorphic structure transform multivariate problem into univariate obtained , according Rouché Theorem inequalities . Conditions by solving the inequality holds , we have come when the fixed base space the dimension N 1 , N 2 with the parameters K , L , can be obtained sufficiently large positive number ( ? ), so that when the fiber dimension n 0 ≥ ( ? ) , obtained Hartogs domains are Lu Qikeng domain. So that we not only give positive examples the series are Lu Qikeng domain , and to determine the conditions provide a basis for further give Lu Qikeng domain geometry .

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CLC: > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Theory of functions > Complex analysis,complex function > Several complex variables function
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