Book Description
This comprehensive monograph is devoted to the study of polynomially convex sets, which play a key role in the theory of functions of several complex variables. "Polynomial Convexity": includes the general properties of polynomially convex sets with particular attention to the theory of the hulls of one-dimensional sets; motivates the theory with numerous examples and counterexamples, which serve to illustrate the general theory and to delineate its boundaries; and examines in considerable detail questions of uniform approximation, especially on totally real sets, for the most part on compact sets but with some attention to questions of global approximation on noncompact sets. It also discusses important applications, e.g., to the study of analytic varieties and to the theory of removable singularities for CR functions; and requires of the reader a solid background in real and complex analysis together with some previous experience with the theory of functions of several complex variables as well as the elements of functional analysis. This beautiful exposition of a rich and complex theory, which contains much material not available in other texts and which is destined to be the standard reference for many years, will appeal to all those with an interest in multivariate complex analysis.
Table of Contents:
Preface.

1           Introduction.

1.1     Polynomial convexity.

1.2     Uniform algebras

1.3     Plurisubharmonic fuctions

1.4     The Cauchy

1.5     Fantappié Integral

1.6     The Oka'Weil Theorem

1.7     Some examples

1.8     Hulls with no analytic structure.

2           Some General Properties of Polynomially Convex Sets

2.1     Applications of the Cousin problems

2.2     Two characterizations of polynomially convex sets

2.3     Applications of Morse theory and algebraic topology

2.4     Convexity in Stein manifolds.

3           Sets of Finite Length

3.1     Introduction

3.2     One-dimensional varieties

3.3     Geometric preliminaries

3.4     Function-theoretic preliminaries

3.5     Subharmonicity results

3.6     Analytic structure in hulls

3.7     Finite area

3.8     The continuation of varieties.

4           Sets of Class A1

4.1     Introductory remarks

4.2     Measure-theoretic preliminaries

4.3     Sets of class A1

4.4     Finite area

4.5     Stokes's Theorem

4.6     The multiplicity function

4.7     Counting the branches.

5           Further Results

5.1     Isoperimetry

5.2     Removable singularities

5.3     Surfaces in strictly pseudoconvex boundaries.

6           Approximation

6.1     Totally real manifolds

6.2     Holomorphically convex sets

6.3     Approximation on totally real manifolds

6.4     Some tools from rational approximation

6.5     Algebras on surfaces

6.6     Tangential approximation.

7           Varieties in Strictly Pseudoconvex Domains

7.1     Interpolation

7.2     Boundary regularity

7.3     Uniqueness.

8           Examples and Counter Examples

8.1     Unions of planes and balls

8.2     Pluripolar graphs

8.3     Deformations

8.4     Sets with symmetry.

References

Index.