Theory of Complex Homogeneous Bounded Domains复齐性有界域理论（英文版）

Theory of Complex Homogeneous Bounded Domains studies the classification and function theory of complex homogeneous bounded domains systematically for the first time. In the book, the Siegel domains are discussed in detail. Proofs are given for
――every homogeneous bounded domain is holomorphically isomorphic to a homogeneous Siegel domain, and
――every homogeneous Siegel domain is affine isomorphic to a normal Siegel domain.
Using the normal Siegel domains to realize the homogeneous bounded domains, we can obtain more property of the geometry and the function theory on homogeneous bounded domains.
This book is suitable for graduate students,and researchers in mathematics.

Theory of Complex Homogeneous Bounded Domains studies the classification and function theory of complex homogeneous bounded domains systematically for the first time. In the book, the Siegel domains are discussed in detail. Proofs are given for
——every homogeneous bounded domain is holomorphically isomorphic to a homogeneous Siegel domain, and
——every homogeneous Siegel domain is affine isomorphic to a normal Siegel domain.
Using the normal Siegel domains to realize the homogeneous bounded domains, we can obtain more property of the geometry and the function theory on homogeneous bounded domains.
This book is suitable for graduate students,and researchers in mathematics.

Preface
Chapter 1. SIEGEL DOMAINS
1. Bounded Domains
1.1 Some Conceptions and Symbols
1.2 Bounded Domains
2. Siegel Domains
3. Holomorphic Automorphism Group of Siegel Domains
Chapter 2. HOMOGENEOUS SIEGEL DOMAINS
1. Homogeneous Bounded Domains
2. Homogeneous Siegel Domains
3. Normal J Lie Algebras
4. J Basis of a Normal J Lie Algebra
Chapter 3. NORMAL SIEGEL DOMAINS
1. Normal Cones and Normal Siegel Domains of First Kind
2. Normal Siegel Domains
3. Decomposable Normal Siegel Domains
4. Bergman Kernel Function of Normal Siegel Domains
Chapter 4. OTHER REALIZATIONS
1. Homogeneous Bounded Domain Realization
2. T Algebra Realization
Chapter 5. AUTOMOROPHISM GROUP
1. Affine Automorphism Group of Normal Cones
2. Affine Automorphism Group of Normal Siegel Domains
3. Holomorphic Automorphism Group
4. Other Results
Chapter 6. CLASSIFICATION OF SQUARE DOMAINS
1. Classification of Two Special Matrix Sets
2. Normal Cones and Dual Normal Cones
3. Classification of Square Cones
4. Classification of Dual Square Cones
5. Classification of Square Domains
6. Classification of General Square Domains
Chapter 7. SYMMETRIC BOUNDED DOMAINS
1. Symmetric Bounded Domains
2. Semisimple J Lie Algebras
3. Classification of Symmetric Siegel Domains
4. Realization of Exceptional Cases
Chapter 8. SZEGO KERNEL AND POISSON KERNEL
1. Cauchy-Szego Integral
2. Formal Poisson Kernel Function
3. Stein-Vagi Conjecture
4. Poisson Integral on Symmetric Siegel Domains
Chapter 9. HOMOGENEOUS BOUNDED DOMAINS
1. Non-Semisimple Effective J Lie Algebras
2. Algebraic J Lie Algebras
3. Realization of Homogeneous Siegel Domains
4. Homogeneous Bounded Domains
5. Main Theorem
References
Index

前言：

In tracing the history of classification theory of homogeneous bounded domains in Cn, one must mention the pioneering work by H. Poincare (ref. [161]). In 1907, he proved that the bicylinder (Ⅰz1Ⅰ < 1, Ⅰz2Ⅰ < 1) and the hypersphere (Ⅰz1Ⅰ2 + Ⅰz2Ⅰ2 < 1) in C2 are not holomorphically equivalent to each other, but they are both connected and simply connected.Therefore, the famous Riemann Theorem in the geometric function theory of one complex variable is not true in C 2. . In 1935, E. Cartan (ref. [15]) gave a complete classification of Hermiti..