Complex Analysis and CR Geometry ( University Lecture Series )

Publication series ：University Lecture Series

Author： Giuseppe Zampieri

Publisher： American Mathematical Society‎

Publication year： 2008

E-ISBN: 9781470421878

P-ISBN(Paperback): 9780821844427

Subject： O175.25 Elliptic Equations

Keyword： Analysis

Language： ENG

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Complex Analysis and CR Geometry

Description

Cauchy-Riemann (CR) geometry is the study of manifolds equipped with a system of CR-type equations. Compared to the early days when the purpose of CR geometry was to supply tools for the analysis of the existence and regularity of solutions to the $\bar\partial$-Neumann problem, it has rapidly acquired a life of its own and has became an important topic in differential geometry and the study of non-linear partial differential equations. A full understanding of modern CR geometry requires knowledge of various topics such as real/complex differential and symplectic geometry, foliation theory, the geometric theory of PDE's, and microlocal analysis. Nowadays, the subject of CR geometry is very rich in results, and the amount of material required to reach competence is daunting to graduate students who wish to learn it. However, the present book does not aim at introducing all the topics of current interest in CR geometry. Instead, an attempt is made to be friendly to the novice by moving, in a fairly relaxed way, from the elements of the theory of holomorphic functions in several complex variables to advanced topics such as extendability of CR functions, analytic discs, their infinitesimal deformations, and their lifts to the cotangent space. The choice of topics provides a good balance between a first exposure to CR geometry and subjects representing current research. Even a seasoned mathematician who wants to contribute to the subject of CR analysis and geometry will find the

Chapter

Cover

Title

Contents

Preface

Chapter 1. Several Complex Variables

1.1. Analytic functions in complex spaces

1.2. Cauchy formula in polydiscs

1.3. Analytic functions and power series

1.4. Subharmonic functions

1.5. Separate analyticity

1.6. Levi forms—continuity principle (I)—Hartogs extension theorem

1.7. Logarithmic supermean of Taylor radius of holomorphic functions—continuity principle (II)—propagation of holomorphic extendibility

1.8. Domains of holomorphy and pseudoconvex domains

1.9. L[sup(2)]-estimates for [omitted] on q-pseudoconvex domains of C[sup(n)]

1.10. Subelliptic estimates for [omitted]

Chapter 2. Real Structures

2.1. Euclidean spaces and their diffeomorphisms

2.2. Integration of vector fields and vector bundles—Frobenius theorem

2.3. Real symplectic spaces—Frobenius-Darboux theorem

2.4. Subelliptic estimates and hypoellipticity of systems of vector fields

2.5. Miscellanea: foliations—orbits

Chapter 3. Real/Complex Structures

3.1. Complex structures—real underlying structures—complexifications

3.2. CR manifolds

3.3. CR functions and CR mappings

3.4. The Levi form of a submanifold M ⊂ C[sup(n)] and an abstract CR structure

3.5. Real/complex symplectic spaces

3.6. Approximation of CR functions by polynomials

3.7. Analytic discs and the extension of CR functions: the "edge of the wedge" theorem, the deformation of discs for manifolds of type 2 and the Levi extension

3.8. Iterated commutators, finite type, Bloom-Grahamnormal form: deformation of discs for manifolds of higher type

3.9. Partial lifts of analytic discs and CR curves

3.10. Defect of analytic discs—deformation of non-defective discs—wedge extension from minimal manifolds

3.11. Propagation of CR extendibility

3.12. Separate real analyticity

Bibliography

Subject Index

Symbols Index

Back Cover

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