Analysis and Geometry in Several Complex Variables

This volume consists of a collection of articles for the proceedings of the 40th Taniguchi Symposium Analysis and Geometry in Several Complex Variables held in Katata, Japan, on June 23-28, 1997. Since the inhomogeneous Cauchy-Riemann equation was introduced in the study of Complex Analysis of Several Variables, there has been strong interaction between Complex Analysis and Real Analysis, in particular, the theory of Partial Differential Equations. Problems in Complex Anal­ ysis stimulate the development of the PDE theory which subsequently can be applied to Complex Analysis. This interaction involves Differen­ tial Geometry, for instance, via the CR structure modeled on the induced structure on the boundary of a complex manifold. Such structures are naturally related to the PDE theory. Differential Geometric formalisms are efficiently used in settling problems in Complex Analysis and the results enrich the theory of Differential Geometry. This volume focuses on the most recent developments in this inter­ action, including links with other fields such as Algebraic Geometry and Theoretical Physics. Written by participants in the Symposium, this vol­ ume treats various aspects of CR geometry and the Bergman kernel/ pro­ jection, together with other major subjects in modern Complex Analysis. We hope that this volume will serve as a resource for all who are interested in the new trends in this area. We would like to express our gratitude to the Taniguchi Foundation for generous financial support and hospitality. We would also like to thank Professor Kiyosi Ito who coordinated the organization of the symposium.

Contenu

The Bergman Kernel and a Theorem of Tian.- Some Involutive Structures in Analysis and Geometry.- The Bergman Kernel in Quantum Mechanics.- WKB and the Periodic Table.- Local Sobolev-Bergman Kernels of Strictly Pseudoconvex Domains.- Quantitative Estimates for Global Regularity.- Pdes Associated to the CR Embedding Theorem.- $$ \bar{\partial } $$ and $$ {\bar{\partial }_b} $$ Problems on Nonsmooth Domains.- A Note on the Closed Rangeness of Vector Bundle Valued Tangential Cauchy-Riemann Operator.- Discrete Groups of Complex Hyperbolic Isometries and Pseudo-Hermitian Structures.- Pseudoconvex Domains in ?n: A Question on the 1-Convex Boundary Points.- Existence and Applications of Analytic Zariski Decompositions.- Segre Polar Correspondence and Double Valued Reflection for General Ellipsoids.- G2-Geometry of Overdetermined Systems of Second Order289.