CR Manifolds and the Tangential Cauchy Riemann Complex provides an elementary introduction to CR manifolds and the tangential Cauchy-Riemann Complex and presents some of the most important recent developments in the field. The first half of the book covers the basic definitions and background material concerning CR manifolds, CR functions, the tangential Cauchy-Riemann Complex and the Levi form.
The second half of the book is devoted to two significant areas of current research. The first area is the holomorphic extension of CR functions. Both the analytic disc approach and the Fourier transform approach to this problem are presented. The second area of research is the integral kernal approach to the solvability of the tangential Cauchy-Riemann Complex. CR Manifolds and the Tangential Cauchy Riemann Complex will interest students and researchers in the field of several complex variable and partial differential equations.
Table of Contents
Part I: Preliminaries 1. Analysis on Euclidean Space 2. Analysis on Manifolds 3. Complexified Vectors and Forms 4. The Frobenius Theorem 5. Distribution Theory 6. Currents Part II: CR Manifolds 7. CR Manifolds 8. The Tangential Cauchy-Riemann Complex 9. CR Functions and Maps 10. The Levi Form 11. The Imbeddability of CR Manifolds 12. Further Results Part III: The Holomorphic Extension of CR Functions 13. An Approximation Theorem 14. The Statement of the CR Extension Theorem 15. The Analytic Disc Technique 16. The Fourier Transform Technique 17. Further Results Part IV: Solvability of the Tangential Cauchy-Riemann Complex 18. Kernel Calculus 19. Fundamental Solutions for the Exterior Derivative and Cauchy-Riemann Operators 20. The Kernels of Henkin 21. Fundamental Solutions for the Tangential Cauchy-Riemann Complex on a Convex Hypersurface 22. A Local Solution to the Tangential Cauchy-Riemann Equations 23. Local Nonsolvability of the Tangential Cauchy-Riemann Complex 24. Further Results