Preface ......................................................... v
Index of Frequently Used Notation .............................. ix
1. INTRODUCTION ................................................. 1
1.1. Polynomial Convexity .................................... 1
1.2. Uniform Algebras ........................................ 5
1.3. Plurisubharmonic Functions ............................. 20
1.4. The Cauchy-Fantappie Integral .......................... 28
1.5. The Oka-Weil Theorem ................................... 39
1.6. Some Examples .......................................... 47
1.7. Hulls with No Analytic Structure ....................... 68
2. SOME GENERAL PROPERTIES OF POLYNOMIALLY CONVEX SETS ......... 71
2.1. Applications of the Cousin Problems .................... 71
2.2. Two Characterizations of Polynomially Convex Sets ...... 83
2.3. Applications of Morse Theory and Algebraic Topology .... 93
2.4. Convexity in Stein Manifolds .......................... 106
3. SETS OF FINITE LENGTH ...................................... 121
3.1. Introduction .......................................... 121
3.2. One-Dimensional Varieties ............................. 123
3.3. Geometric Preliminaries ............................... 125
3.4. Function-Theoretic Preliminaries ...................... 132
3.5. Subharmonicity Results ................................ 143
3.6. Analytic Structure in Hulls ........................... 148
3.7. Finite Area ........................................... 152
3.8. The Continuation of Varieties ......................... 156
4. SETS OF CLASS A1 ........................................... 169
4.1. Introductory Remarks .................................. 169
4.2. Measure-Theoretic Preliminaries ....................... 170
4.3. Sets of Class A1 ...................................... 172
4.1. Finite Area ........................................... 181
4.5. Stokes's Theorem ...................................... 183
4.6. The Multiplicity Function ............................. 203
4.7. Counting the Branches ................................. 213
5. FURTHER RESULTS ............................................ 217
5.1. Isoperimetry .......................................... 217
5.2. Removable Singularities ............................... 231
5.3. Surfaces in Strictly Pseudoconvex Boundaries .......... 257
6. APPROXIMATION .............................................. 277
6.1. Totally Real Manifolds ................................ 277
6.2. Holomorphically Convex Sets ........................... 289
6.3. Approximation on Totally Real Manifolds ............... 300
6.4. Some Tools from Rational Approximation ................ 310
6.5. Algebras on Surfaces .................................. 314
6.6. Tangential Approximation .............................. 341
7. VARIETIES IN STRICTLY PSEUDOCONVEX DOMAINS ................. 351
7.1. Interpolation ......................................... 351
7.2. Boundary Regularity ................................... 365
7.3. Uniqueness ............................................ 371
8. EXAMPLES AND COUNTEREXAMPLES ............................... 377
8.1. Unions of Planes and Balls ............................ 377
8.2. Pluripolar Graphs ..................................... 394
8.3. Deformations .......................................... 399
8.4. Sets with Symmetry .................................... 402
References .................................................... 415
Index ......................................................... 431
| 
