Geometric function theory in several complex variables: proceedings of a Satellite Conference to the International Congress of Mathematicians in Beijing 2002, University of Science and Technology, China, 30 August - 2 September 2002

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. Contents......Page 8 Preface......Page 6 Subriemannian geometry and subelliptic partial differential equations (by Der-Chen Chang, Peter C. Greiner and Jingzhi Tie)......Page 10 1. Euclidean Laplacian and elliptic operators......Page 12 2. The Heisenberg group and the sub-Laplacian......Page 15 3. The Hamilton-Jacobi equation and the heat kernel......Page 26 4. SubRiemann geometry associated to step 3 sub-Laplacian......Page 35 References......Page 44 1. Preface......Page 46 2. Re-explanation of ideas of M-L-L-0's Algorithm......Page 50 3. Estimation of time-space in computing defective sum......Page 53 4. Some Remarks......Page 55 References......Page 58 1. Introduction......Page 59 2. Hardy space of infinite complex variables......Page 60 3. Multipliers and the N-shift......Page 68 4. von Neumann’s inequality......Page 72 References......Page 74 1. Introduction......Page 75 2. Preliminaries......Page 77 3. Proof of Theorem......Page 80 References......Page 82 1. Introduction......Page 83 2. Preliminary......Page 84 3. The proof of theorem 1.2......Page 89 References......Page 90 0. Introduction......Page 91 1. General description of cones G(B) and G[T]......Page 95 2. Generators of one-dimensional type......Page 105 3. Differential equations for starlike and spirallike mappings in H = Cn......Page 110 References......Page 124 1. One Variable Invariant Functions......Page 127 2. Several Variable Invariant Mappings......Page 129 4. Acknowledgement......Page 131 1. Introduction......Page 132 2. Green-Goursat......Page 136 3. Approximation......Page 138 4. Natural domains......Page 140 5. Nonrectifiable boundary......Page 146 6. Discontinuous boundary functions......Page 149 References......Page 150 1. Introduction......Page 152 2. The estimate of Jf(z)Jf(z) '.......Page 153 3. Distortion theorem for linear invariant family......Page 155 4. Distortion theorem for bounded symmetric domains......Page 157 References......Page 158 1. Introduction and results......Page 160 2. Estimates for periodic points of a special family of holomorphic maps......Page 162 3. Formally linearizable maps......Page 170 References......Page 173 1. Introduction and preliminaries......Page 174 2. The generalization of the Caratheodory class......Page 177 3. Loewner chains and the Loewner differential equation......Page 180 4. Lipschitz continuity and its consequences......Page 183 5. The Roper-Suffridge extension operator......Page 186 References......Page 188 1. Introduction......Page 191 2. The Euler-Lagrange Cohomology Group of Degree 1......Page 192 2.1. The Euler-Lagrange 1-Forms in the Lagrange Mechanics......Page 193 2.2. The Euler-Lagmnge I-Form on a Symplectic Manifold......Page 195 2.3. The Euler-Lagrange Cohomology Group of Degree 1......Page 197 3. The Euler-Lagrange Cohomology Groups on Symplectic Manifolds......Page 200 3.1. The Euler-Lagrangian Cohomology Group of Degree 2k – 1......Page 201 3.2. Some Operators......Page 202 3.3. The Spaces Xik-1 ( M , w ) and H (M, u)......Page 204 3.4. The Other Euler-Lagrange Cohomology Groups......Page 205 3.5. Euler-Lagrange Cohomology and Harmonic Cohomology......Page 208 3.6. The Relative Euler-Lagrange Cohomology......Page 209 4.1. The Derivation of the Equations......Page 210 4.2. On The Canonical Hamiltonian Equations, The l'race of 2-Forms and The Poisson Bracket......Page 212 5. Discussions and Conclusions......Page 214 Acknowledgement......Page 215 References......Page 216 1. A new inequality......Page 217 2. Some applications of the Theorem 1......Page 219 References......Page 220 1. Introduction......Page 221 2. Main Theorems......Page 223 References......Page 227 1. Introduction and main theorems......Page 229 2. The Proof of Theorem 1.3......Page 232 3. The proof of Theorem 1.1......Page 238 4. The Proof of Theorem 1.4......Page 242 5. Proof of Proposition 1.2......Page 251 References......Page 252 1. Introduction......Page 254 2. Definitions and main theorems......Page 255 Acknowledgments......Page 259 References......Page 260 1. Introduction......Page 261 2. Preliminaries......Page 262 3. Main Theorems and Proof......Page 267 References......Page 273 The growth and 1/2-covering theorems for quasi-convex mappings (by Taishun Liu and Wenjun Zhang)......Page 274 1. Quasi-convex mapping of type A and Quasi-convex mappings in Complex Banach Space......Page 276 2. Several Lemmas......Page 282 3. The Growth and Covering Theorems of Quasi-Convex Mappings......Page 284 References......Page 287 1. Introduction......Page 288 2. Main theorem......Page 289 3. Peano curve method......Page 291 4. Examples......Page 292 References......Page 295 1. Introduction......Page 296 2. Complex Finsler manifolds and invariant integral kernel......Page 297 3. Invariant integral kernel in local coordinates......Page 300 4. The Koppelman formula for differential forms of type (P, 4)......Page 301 References......Page 303 1. Fundamental Concepts......Page 304 2. Extension to the Boundary......Page 307 3. Convex Domains that Contain A Line......Page 309 4. Other Circular Domains in Cn......Page 312 5. Open Problems......Page 316 References......Page 317 Rigidity of proper holomorphic mappings between bounded symmetric domains (by Zhen-Han Tu)......Page 319 Acknowledgments......Page 323 References......Page 324 1. Preliminary......Page 326 2. Hardamard Three-Circle Theorem on Riemann Surface......Page 330 References......Page 333 1. Introduction......Page 334 2.1. Case I: If ( M, F ) is a compact Hermitian manifold......Page 335 2.2. Case II: If ( M , F ) is a strongly pseudoconvex compact complex Finsler manifold......Page 336 3. Hermitian product on projectivized tangent bundle PTM......Page 337 4. Global Hermitian inner product and Hodge-Laplace operator on compact complex Finsler manifolds......Page 338 References......Page 341 1. Introduction......Page 342 2. Some Lemmas......Page 344 3. The Proof of Theorem 1......Page 345 4. The Proof of Theorem 2......Page 347 References......Page 350 "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."--BOOK JACKET