**University Lecture Series**

Volume: 43;
2008;
200 pp;
Softcover

MSC: Primary 32;

Print ISBN: 978-0-8218-4442-7

Product Code: ULECT/43

List Price: $49.00

Individual Member Price: $39.20

**Electronic ISBN: 978-1-4704-2187-8
Product Code: ULECT/43.E**

List Price: $49.00

Individual Member Price: $39.20

#### Supplemental Materials

# Complex Analysis and CR Geometry

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*Giuseppe Zampieri*

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 choice of topics attractive.

#### Table of Contents

# Table of Contents

## Complex Analysis and CR Geometry

- Cover Cover11 free
- Title i2 free
- Copyright ii3 free
- Contents v6 free
- Preface vii8 free
- Chapter 1. Several Complex Variables 110 free
- 1.1. Analytic functions in complex spaces 110
- 1.2. Cauchy formula in polydiscs 514
- 1.3. Analytic functions and power series 1120
- 1.4. Subharmonic functions 1423
- 1.5. Separate analyticity 2231
- 1.6. Levi forms—continuity principle (I)—Hartogs extension theorem 2433
- 1.7. Logarithmic supermean of Taylor radius of holomorphic functions—continuity principle (II)—propagation of holomorphic extendibility 3039
- 1.8. Domains of holomorphy and pseudoconvex domains 3948
- 1.9. L[sup(2)]-estimates for [omitted] on q-pseudoconvex domains of C[sup(n)] 5059
- 1.10. Subelliptic estimates for [omitted] 6776

- Chapter 2. Real Structures 7786
- 2.1. Euclidean spaces and their diffeomorphisms 7786
- 2.2. Integration of vector fields and vector bundles—Frobenius theorem 8392
- 2.3. Real symplectic spaces—Frobenius-Darboux theorem 9099
- 2.4. Subelliptic estimates and hypoellipticity of systems of vector fields 96105
- 2.5. Miscellanea: foliations—orbits 100109

- Chapter 3. Real/Complex Structures 107116
- 3.1. Complex structures—real underlying structures—complexifications 107116
- 3.2. CR manifolds 113122
- 3.3. CR functions and CR mappings 123132
- 3.4. The Levi form of a submanifold M ⊂ C[sup(n)] and an abstract CR structure 132141
- 3.5. Real/complex symplectic spaces 136145
- 3.6. Approximation of CR functions by polynomials 144153
- 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 151160
- 3.8. Iterated commutators, finite type, Bloom-Grahamnormal form: deformation of discs for manifolds of higher type 159168
- 3.9. Partial lifts of analytic discs and CR curves 168177
- 3.10. Defect of analytic discs—deformation of non-defective discs—wedge extension from minimal manifolds 174183
- 3.11. Propagation of CR extendibility 177186
- 3.12. Separate real analyticity 183192

- Bibliography 191200
- Subject Index 197206
- Symbols Index 199208
- Back Cover Back Cover1210

#### Readership

Graduate students and research mathematicians interested in complex analysis and differential geometry.

#### Reviews

One nice feature of the book is the “Suggested research” sections in which the author discusses open problems related to the material of the chapter. It also has guided exercises, in which the reader is encouraged to provide proofs for certain technical aspects following a given recipe or a hint.

-- Mathematical Reviews