Complex Analysis - Several Complex Variables and Connections with PDE Theory, and Geometry

Cover......Page 1
Complex Analysis......Page 4
ISBN 978-3-0346-0008-8......Page 5
Contents......Page 6
Preface......Page 8
The subject......Page 9
Organization......Page 10
Extended Curriculum Vitae of Linda Preiss Rothschild......Page 12
Honors and Fellowships......Page 13
Selected National Committees and Offices......Page 14
Editorial Positions......Page 15
Publication List of Linda Preiss Rothschild......Page 16
Introduction......Page 23
1. Polar structure of X |f|2λ ......Page 25
2. Existence of polar oblique lines......Page 27
3. Pullback and interaction......Page 31
4. Interaction of strata revised......Page 33
5. Examples......Page 40
References......Page 44
0. Introduction......Page 47
1. Preliminaries......Page 48
2. Main results and examples......Page 52
3. Some lemmas and the proof of Theorem 2.1......Page 59
4. Proofs of Theorem 2.4 and Theorem 2.7......Page 66
References......Page 71
1. Introduction......Page 73
2. The operator P is C∞ hypoelliptic......Page 75
3. Gevrey hypoellipticity......Page 78
4.1. q-pseudodifferential calculus......Page 80
4.2. The actual computation of the eigenvalue......Page 86
4.3. Hypoellipticity of P......Page 91
A. Appendix......Page 93
References......Page 94
1. Introduction......Page 97
2. Definition of subelliptic estimates......Page 98
3. Subelliptic estimates in two dimensions......Page 99
4. Subelliptic multipliers......Page 101
5. Triangular systems......Page 104
6. Necessary and sufficient conditions for subellipticity......Page 110
7. Sharp subelliptic estimates......Page 112
References......Page 114
1. Introduction......Page 117
2. Properties of the invariant polynomials......Page 120
3. Cyclic groups......Page 121
4. Asymptotic information......Page 123
5. Metacyclic groups......Page 126
6. An application; failure of rigidity......Page 127
References......Page 129
On the Subellipticity of Some Hypoelliptic Quasihomogeneous Systems of Complex Vector Fields......Page 131
1.1. Preliminaries on subellipticity and hypoellipticity......Page 132
1.2. The main results......Page 133
1.3. Comparison with previous results......Page 134
2. Derridj’s subellipticity criterion......Page 135
3.1. Distorted geometry......Page 136
3.2. Distorted dynamics......Page 137
4.3. The lower bound in the quasi-homogeneous case......Page 139
4.4. The case of arcs in ϕ ≥ 0 but with a zero at one end......Page 141
5. Completion of the proof......Page 142
Appendix A. A technical proposition......Page 143
References......Page 144
1. Oka properties......Page 147
2. Subelliptic submersions and Serre fibrations......Page 151
4. A parametric Oka principle for liftings......Page 154
5. Ascent and descent of the parametric Oka property......Page 163
References......Page 164
1. Introduction......Page 167
2. Preliminaries......Page 168
3. Positivity of the spectrum and essential spectrum......Page 172
4. Hearing pseudoconvexity......Page 174
References......Page 178
1. Introduction......Page 181
2. Weighted basic estimates......Page 183
3. Weighted Sobolev spaces......Page 187
4. Compactness estimates......Page 190
References......Page 194
Remarks on the Homogeneous Complex Monge-Ampere Equation......Page 197
References......Page 206
1. Introduction......Page 209
2. The approximation theorem......Page 210
3. Structures of co-rank one......Page 216
4. A Rado theorem for structures of co-rank one......Page 217
5. An application to uniqueness......Page 222
References......Page 224
1. Introduction......Page 227
2. The parametric Oka principle for liftings......Page 229
3. Equivalence of the basic and the parametric Oka properties......Page 230
4. The convex interpolation property......Page 231
References......Page 233
Stability of the Vanishing of the ∂b-cohomology Under Small Horizontal Perturbations of the CR Structure in Compact Abstract q-concave CR Manifolds......Page 235
1. CR structures......Page 237
2. Stability of vanishing theorems by horizontal perturbations of the CR structure......Page 239
3. Anisotropic spaces......Page 243
References......Page 247
1. Introduction......Page 249
2. Cohesive sheaves, an overview......Page 250
3. Tensor products......Page 253
5. Preparation......Page 256
6. Hom and ⊗......Page 260
8. Application. Complex analytic subspaces and subvarieties......Page 264
References......Page 266
1. Introduction......Page 267
2. Complex b-manifolds......Page 269
3. Classification by relative Chern classes......Page 271
4. Classification by a Picard group......Page 279
5. The bundle V and the orbits of the structure group G......Page 280
References......Page 284
Introduction......Page 285
1. A singular CR equation......Page 286
2. Normal form of a class of vector fields......Page 288
3. Solvability for the vector field X......Page 289
4. A semilinear equation......Page 291
5. Equations for the bending fields......Page 293
6. Local non rigidity of a class of surfaces......Page 296
References......Page 299
On the Zariski Closure of a Germ of Totally Geodesic Complex Submanifold on a Subvariety of a Complex Hyperbolic Space Form of Finite Volume......Page 301
1. Statement of the main result and background materials......Page 303
2. Proof of the results......Page 311
References......Page 321
The Large Time Asymptotics of the Entropy......Page 323
References......Page 327
1. Introduction......Page 329
2. The ¯ ∂-equation on the annulus between two strictly pseudoconvex domains in Cn......Page 330
3. The ¯ ∂-equation on the annulus between two weakly pseudoconvex domains in Cn......Page 334
4. The ¯ ∂-equation on weakly pseudoconcave domains in CPn......Page 338
References......Page 341
1. Introduction......Page 343
2. Transformation rule and its expansion......Page 347
3. Partial normalization in general case......Page 348
4. Levi-nondegenerate case......Page 349
5. Weight estimates......Page 351
6. Trace decompositions......Page 352
7.1. Normalization for k ≥ 2......Page 356
7.2. Normalization for k = 1......Page 357
7.3. Normalization for k = 0......Page 360
References......Page 361