Geometric Function Theory in Several Complex Variables: Proceedings of a Satellite Conference to the International Congress on Mathematicians in Beijing 2002, University of Science and Technology, China, 30 August - 2 September 2002

✍ Scribed by Carl Hanson FitzGerald; Sheng Gong

• Publisher: World Scientific
• Year: 2004
• Tongue: English
• Leaves: 352
• Category: Library

✦ Synopsis

The papers contained in this book address problems in one and several complex variables. The main theme is the extension of geometric function theory methods and theorems to several complex variables. The papers present various results on the growth of mappings in various classes as well as observations about the boundary behavior of mappings, via developing and using some semi group methods.

✦ Table of Contents

Contents
Preface
Subriemannian geometry and subelliptic partial differential equations (by Der-Chen Chang, Peter C. Greiner and Jingzhi Tie)
1. Euclidean Laplacian and elliptic operators
2. The Heisenberg group and the sub-Laplacian
3. The Hamilton-Jacobi equation and the heat kernel
4. SubRiemann geometry associated to step 3 sub-Laplacian
Acknowledgments
References
Defective values of double Meissel’s formula and reduction of space-time requirement of Meissel–Lehmer–Lagarias–Odlyzko’s algorithm an experimental program to find T(1021) (by Chen Guangxiao)
1. Preface
2. Re-explanation of ideas of M-L-L-0's Algorithm
3. Estimation of time-space in computing defective sum
4. Some Remarks
References
Hardy space of holomorphic functions in infinite complex variables (by Zeqian Chen)
1. Introduction
2. Hardy space of infinite complex variables
3. Multipliers and the N-shift
4. von Neumann’s inequality
References
The law of the iterated logarithm for pluriharmonic functions in the unit ball of Cn (by Zeqian Chen, Caiheng Ouyang)
1. Introduction
2. Preliminaries
3. Proof of Theorem
Acknowledgement
References
Proper holomorphic mappings between some generalized Hartogs triangles (by Zhihua Chen)
1. Introduction
2. Preliminary
3. The proof of theorem 1.2
References
Semigroups of holomorphic mappings with boundary fixed points and spirallike mappings (by Mark Elin and David Shoikhet)
0. Introduction
1. General description of cones G(B) and G[T]
2. Generators of one-dimensional type
3. Differential equations for starlike and spirallike mappings in H = Cn
Acknowledgments
References
Invariant mappings in geometric function theory (by Carl H. FitzGerald)
1. One Variable Invariant Functions
2. Several Variable Invariant Mappings
3. Conclusion
4. Acknowledgement
The Cauchy Theorem for domains of arbitrary connectivity in Ftiemann surfaces (by P. M. Gauthier)
1. Introduction
2. Green-Goursat
3. Approximation
4. Natural domains
5. Nonrectifiable boundary
6. Discontinuous boundary functions
References
The distortion theorems for convex mappings in several complex variables (by Sheng Gong)
1. Introduction
2. The estimate of Jf(z)Jf(z) '.
3. Distortion theorem for linear invariant family
4. Distortion theorem for bounded symmetric domains
References
Anti-holomorphiclly reversible holomorphic maps that are not holomorphically reversible (by Xianghong Gong)
1. Introduction and results
2. Estimates for periodic points of a special family of holomorphic maps
3. Formally linearizable maps
References
Basic properties of Loewner chains in several complex variables (by Ian Graham, Gabriela and Mirela Kohr)
1. Introduction and preliminaries
2. The generalization of the Caratheodory class
3. Loewner chains and the Loewner differential equation
4. Lipschitz continuity and its consequences
5. The Roper-Suffridge extension operator
Acknowledgments
References
The Euler-Lagrange cohomology on symplectic manifolds (by Han-Ying Guo, Jianzhong Pan, Ke Wu and Bin Zhou)
1. Introduction
2. The Euler-Lagrange Cohomology Group of Degree 1
2.1. The Euler-Lagrange 1-Forms in the Lagrange Mechanics
2.2. The Euler-Lagmnge I-Form on a Symplectic Manifold
2.3. The Euler-Lagrange Cohomology Group of Degree 1
3. The Euler-Lagrange Cohomology Groups on Symplectic Manifolds
3.1. The Euler-Lagrangian Cohomology Group of Degree 2k – 1
3.2. Some Operators
3.3. The Spaces Xik-1 ( M , w ) and H (M, u)
3.4. The Other Euler-Lagrange Cohomology Groups
3.5. Euler-Lagrange Cohomology and Harmonic Cohomology
3.6. The Relative Euler-Lagrange Cohomology
4. The General Volume-Preserving Hamiltonian Equations
4.1. The Derivation of the Equations
4.2. On The Canonical Hamiltonian Equations, The l'race of 2-Forms and The Poisson Bracket
4.3. One Possible Application
5. Discussions and Conclusions
Acknowledgement
References
A new inequality and its applications (by Hu Ke)
1. A new inequality
2. Some applications of the Theorem 1
References
Extended Cesbro operators on the Bloch space in the unit ball of Cn (by Hu, Zhangjian)
1. Introduction
2. Main Theorems
References
On the criteria for Schatten von Neumann class of composition operators on Hardy and Bergman spaces in domains in Cn (by Song-Ying Li)
1. Introduction and main theorems
2. The Proof of Theorem 1.3
3. The proof of Theorem 1.1
4. The Proof of Theorem 1.4
5. Proof of Proposition 1.2
References
The higher order linear partial differential integral equations on closed smooth manifolds in Cn (by Liangyu Lin, Chunhui Qiu and Yusheng Huang)
1. Introduction
2. Definitions and main theorems
Acknowledgments
References
The new characteristics for spirallike mappings of type a on bounded balanced pseudoconvex domains (by Hao Liu)
1. Introduction
2. Preliminaries
3. Main Theorems and Proof
References
The growth and 1/2-covering theorems for quasi-convex mappings (by Taishun Liu and Wenjun Zhang)
1. Quasi-convex mapping of type A and Quasi-convex mappings in Complex Banach Space
2. Several Lemmas
3. The Growth and Covering Theorems of Quasi-Convex Mappings
References
Intermediate value theorem for functions of classes of Riemann surfaces (by Makoto Masumoto)
1. Introduction
2. Main theorem
3. Peano curve method
4. Examples
References
Integral formula for differential forms of type (P, Q) on complex Finsler manifolds (by Chunhui Qui and Tongde Zhong)
1. Introduction
2. Complex Finsler manifolds and invariant integral kernel
3. Invariant integral kernel in local coordinates
4. The Koppelman formula for differential forms of type (P, 4)
Acknowledgments
References
Holomorphic mappings of domains in Cn onto convex domains (by Ted J. Suffridge)
1. Fundamental Concepts
2. Extension to the Boundary
3. Convex Domains that Contain A Line
4. Other Circular Domains in Cn
5. Open Problems
References
Rigidity of proper holomorphic mappings between bounded symmetric domains (by Zhen-Han Tu)
Acknowledgments
References
A Hadamard theorem on algebraic curves (by Shi-Kun Wang and Hui-Ping Zhang)
1. Preliminary
2. Hardamard Three-Circle Theorem on Riemann Surface
Acknowledgement
References
Hodge-Laplace operator on complex Finsler manifolds (by Chunping Zhong and Tongde Zhong)
1. Introduction
2. Complex Finsler metric on complex manifold
2.1. Case I: If ( M, F ) is a compact Hermitian manifold
2.2. Case II: If ( M , F ) is a strongly pseudoconvex compact complex Finsler manifold
3. Hermitian product on projectivized tangent bundle PTM
4. Global Hermitian inner product and Hodge-Laplace operator on compact complex Finsler manifolds
Acknowledgments
References
Weighted composition operators on the Lipschitz space in polydiscs (by Zehua Zhou)
1. Introduction
2. Some Lemmas
3. The Proof of Theorem 1
4. The Proof of Theorem 2
References