Complex Analysis (eBook)

Title Page 3
Copyright Page 4
Table of Contents 5
Preface 7
Focus on youth 8
The subject 8
Organization 9
Extended Curriculum Vitae of Linda Preiss Rothschild 11
Educational Background 12
Professional Employment 12
Honors and Fellowships 12
Selected Invited Lectures 13
Students 13
Selected National Committees and Offices 13
Editorial Positions 14
Publication List of Linda Preiss Rothschild 15
Oblique Polar Lines of X |f|2.|g|2µ 21
Introduction 21
1. Polar structure of X |f|2. 23
2. Existence of polar oblique lines 25
3. Pullback and interaction 29
4. Interaction of strata revised 31
5. Examples 38
References 42
On Involutive Systems of First-order Nonlinear Partial Differential Equations 44
0. Introduction 44
1. Preliminaries 45
2. Main results and examples 49
3. Some lemmas and the proof of Theorem 2.1 56
4. Proofs of Theorem 2.4 and Theorem 2.7 63
References 68
Gevrey Hypoellipticity for an Interesting Variant of Kohn’s Operator 70
1. Introduction 70
2. The operator P is C8 hypoelliptic 72
3. Gevrey hypoellipticity 75
4. Computing . 77
4.1. q-pseudodifferential calculus 77
4.2. The actual computation of the eigenvalue 83
4.3. Hypoellipticity of P 88
A. Appendix 90
References 91
Subelliptic Estimates 93
1. Introduction 93
2. Definition of subelliptic estimates 94
3. Subelliptic estimates in two dimensions 95
4. Subelliptic multipliers 97
5. Triangular systems 100
6. Necessary and sufficient conditions for subellipticity 106
7. Sharp subelliptic estimates 108
References 110
Invariant CR Mappings 113
1. Introduction 113
2. Properties of the invariant polynomials 116
3. Cyclic groups 117
4. Asymptotic information 119
5. Metacyclic groups 122
6. An application failure of rigidity123
References 125
On the Subellipticity of Some Hypoelliptic Quasihomogeneous Systems of Complex Vector Fields 126
1. Introduction and main result 127
1.1. Preliminaries on subellipticity and hypoellipticity 127
1.2. The main results 128
1.3. Comparison with previous results 129
2. Derridj’s subellipticity criterion 130
3. Quasihomogeneous structure 131
3.1. Distorted geometry 131
3.2. Distorted dynamics 132
4. Analysis of the quasielliptic case ( . > 0)
4.1. Construction of . 134
4.2. Analysis of .(t)m - .m 134
4.3. The lower bound in the quasi-homogeneous case 134
4.4. The case of arcs in . = 0 but with a zero at one end 136
5. Completion of the proof 137
Appendix A. A technical proposition 138
References 139
Invariance of the Parametric Oka Property 141
1. Oka properties 141
2. Subelliptic submersions and Serre fibrations 145
3. Convex approximation property 148
4. A parametric Oka principle for liftings 148
5. Ascent and descent of the parametric Oka property 157
References 158
Positivity of the ¯ .-Neumann Laplacian 161
1. Introduction 161
2. Preliminaries 162
3. Positivity of the spectrum and essential spectrum 166
4. Hearing pseudoconvexity 168
References 172
Compactness Estimates for the .-Neumann Problem in Weighted L2-spaces 175
1. Introduction 175
2. Weighted basic estimates 177
3. Weighted Sobolev spaces 181
4. Compactness estimates 184
References 188
Remarks on the Homogeneous Complex Monge-Ampere Equation 191
References 200
A Rado Theorem for Locally Solvable Structures of Co-rank One 202
1. Introduction 202
2. The approximation theorem 203
3. Structures of co-rank one 209
4. A Rado theorem for structures of co-rank one 210
5. An application to uniqueness 215
References 217
Applications of a Parametric Oka Principle for Liftings 219
1. Introduction 219
2. The parametric Oka principle for liftings 221
3. Equivalence of the basic and the parametric Oka properties 222
4. The convex interpolation property 223
References 225
Stability of the Vanishing of the .b-cohomology Under Small Horizontal Perturbations of the CR Structure in Compact Abstract q-concave CR Manifolds 226
1. CR structures 228
2. Stability of vanishing theorems by horizontal perturbations of the CR structure 230
3. Anisotropic spaces 234
References 238
Coherent Sheaves and Cohesive Sheaves 239
1. Introduction 239
2. Cohesive sheaves, an overview 240
3. Tensor products 243
4. The main results 246
5. Preparation 246
6. Hom and . 250
7. Coherence and cohesion 254
8. Application. Complex analytic subspaces and subvarieties 254
References 256
Characteristic Classes of the Boundary of a Complex b-manifold 257
1. Introduction 257
2. Complex b-manifolds 259
3. Classification by relative Chern classes 261
4. Classification by a Picard group 269
5. The bundle V and the orbits of the structure group G 270
References 274
Solvability of Planar Complex Vector Fields with Applications to Deformation of Surfaces 275
Introduction 275
1. A singular CR equation 276
2. Normal form of a class of vector fields 278
3. Solvability for the vector field X 279
4. A semilinear equation 281
5. Equations for the bending fields 283
6. Local non rigidity of a class of surfaces 286
References 289
On the Zariski Closure of a Germ of Totally Geodesic Complex Submanifold on a Subvariety of a Complex Hyperbolic Space Form of Finite Volume 291
1. Statement of the main result and background materials 293
2. Proof of the results 301
References 311
The Large Time Asymptotics of the Entropy 313
References 317
The Closed Range Property for . on Domains with Pseudoconcave Boundary 319
1. Introduction 319
2. The ¯ .-equation on the annulus between two strictly pseudoconvex domains in Cn 320
3. The ¯ .-equation on the annulus between two weakly pseudoconvex domains in Cn 324
4. The ¯ .-equation on weakly pseudoconcave domains in CPn 328
References 331
New Normal Forms for Levi-nondegenerate Hypersurfaces 333
1. Introduction 333
2. Transformation rule and its expansion 337
3. Partial normalization in general case 338
4. Levi-nondegenerate case 339
5. Weight estimates 341
6. Trace decompositions 342
7. Normalizations 346
7.1. Normalization for k = 2 346
7.2. Normalization for k = 1 347
7.3. Normalization for k = 0 350
References 351