Aspects Of Boundary Problems In Analysis And Geometry

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ISBN: 9783764370695 | 3764370696
Cover: Hardcover
Copyright: 5/14/2004

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Boundary problems constitute an essential field of common mathematical interest. The intention of this volume is to highlight several analytic and geometric aspects of boundary problems with special emphasis on their interplay. It includes surveys on classical topics presented from a modern perspective as well as reports on current research. The collection splits into two related groups: - analysis and geometry of geometric operators and their index theory - elliptic theory of boundary value problems and the Shapiro-Lopatinsky condition

Preface

Part I. Geometric Operators and the Index

Spectral invariants of operators of Dirac type on partitioned manifolds

by DAVID BLEECKER AND BERNHELM BOOSS-BAVNBEK

Introduction

(130)

1. Basic notations and results

(25)

1.1. Index of Fredholm operators and spectral flow of curves of self-adjoint Fredholm operators

(9)

1.2. Symmetric operators and symplectic analysis

(7)

1.3. Operators of Dirac type and their ellipticity

(1)

1.4. Weak unique continuation property

(8)

2. The index of elliptic operators on partitioned manifolds

(51)

2.1. Examples and the Hellwig-Vekua index theorem

(10)

2.2. The index of twisted Dirac operators on closed manifolds

(14)

2.3. Dirac type operators on manifolds with boundary

(10)

2.4. The Atiyah-Patodi-Singer index theorem

(11)

2.5. Symplectic geometry of Cauchy data spaces

(3)

2.6. Non-additivity of the index

(2)

2.7. Pasting of spectral flow

(1)

3. The eta invariant

(47)

3.1. Functional integrals and spectral asymmetry

(3)

3.2. The ζ-determinant for operators of infinite rank

(2)

3.3. Spectral invariants of different 'sensitivity'

(2)

3.4. Pasting formulas for the eta invariant - outlines

(6)

3.5. The adiabatic additivity of the small-t chopped η-invariant

(7)

3.6. Asymptotic vanishing of the large-t chopped η-invariant on stretched part manifold

103

(6)

3.7. The estimate of the lowest nontrivial eigenvalue

109

(6)

3.8. The spectrum on the closed stretched manifold

115

(7)

3.9. The additivity for spectral boundary conditions

122

(4)

References

126

(5)

Index theory of Dirac operators on manifolds with corners up to codimension two

by PAUL LOYA

1. Introduction: The Gauss-Bonnet formula and index theory

131

(4)

1.1. The classical Gauss-Bonnet formula

132

(1)

1.2. The Gauss-Bonnet formula as an index formula

133

(2)

2. The Atiyah-Singer index formula

135

(5)

2.1. Statement of the Atiyah-Singer index theorem

135

(3)

2.2. Outline of the proof of the Atiyah-Singer formula

138

(2)

2.3. Some remarks on the Atiyah-Singer index theorem

140

(1)

3. The Atiyah-Patodi-Singer index formula

140

(8)

3.1. Attaching a cylindrical end

141

(3)

3.2. Statement of the Atiyah-Patodi-Singer index theorem

144

(2)

3.3. Interpretation as b-objects

146

(1)

3.4. Some remarks on the Atiyah-Patodi-Singer index theorem

147

(1)

4. Melrose's b-geometry proof of the Atiyah-Patodi-Singer theorem

148

(5)

4.1. The proof of APS with details left out

148

(2)

4.2. Some facts about the heat kernels

150

(1)

4.3. Filling in the details for the b-trace

150

(2)

4.4. Filling in the details for the eta invariant

152

(1)

5. Index theory on manifolds with corners of codimension two

153

(8)

5.1. Dirac operators on manifolds with corners

154

(2)

5.2. Attaching multi-cylindrical ends

156

(2)

5.3. Muller's generalization of the APS index formula

158

(1)

5.4. b-version of Muller's theorem

159

(1)

5.5. Some remarks on index theory on manifolds with corners

160

(1)

6. Perturbations of Dirac operators on manifolds with corners

161

(5)

6.1. Fredholm perturbation of Dirac operators

161

(2)

6.2. An index formula for perturbed Dirac operators

163

(2)

6.3. Some concluding remarks

165

(1)

References

166

(4)

Index defects in the theory of spectral boundary value problems

by ANTON SAVIN AND BORIS STERNIN

Introduction

170

(12)

1. Spectral boundary value problems and their index

182

(8)

1.1. Atiyah-Patodi-Singer spectral boundary value problems

182

(1)

1.2. The spectral flow

183

(3)

1.3. A theorem on index decompositions

186

(3)

1.4. Examples

189

(1)

2. Index defects for problems with parity conditions

190

(8)

2.1. The dimension functional for even subspaces

190

(3)

2.2. The index defect formula

193

(3)

2.3. The dimension functional and the η-invariant

196

(2)

3. Index defects on twisted Zn-manifolds

198

(14)

3.1. Twisted Zn-manifolds and elliptic operators

198

(3)

3.2. The Freed-Melrose index theorem modulo n

201

(2)

3.3. The index defect problem on twisted Zn-manifolds

203

(2)

3.4. The element of K-theory defined by a manifold whose boundary is a covering

205

(2)

3.5. The index defect formula

207

(4)

3.6. An application to η-invariants

211

(1)

4. Appendix A. The Atiyah-Patodi-Singer η-invariant

212

(5)

4.1. The geometric index formula and the η-invariant

212

(2)

4.2. The derivative of the η-invariant

214

(1)

4.3. The homotopy invariance of the η-invariant

214

(3)

5. Appendix B. Elliptic operators and Poincaré duality. Smooth theory

217

(8)

5.1. The Poincaré isomorphism on a closed manifold

218

(4)

5.2. Duality and the topological index

222

(1)

5.3. Poincaré duality on manifolds with boundary. Absolute and relative cycles

223

(2)

6. Appendix C. Poincaré duality on Zn-manifolds

225

(10)

6.1. Relative cycles. The C*-algebra of a Zn-manifold

226

(2)

6.2. Absolute cycles. Nonlocal operators

228

(3)

6.3. The Poincaré isomorphism

231

(1)

6.4. Poincaré duality

232

(2)

6.5. A topological index for Zn-manifolds

234

(1)

References

235

(4)

Cyclic homology and pseudodifferential operators, a survey

by MOULAY-TAHAR BENAMEUR, JACEK BRODZKI, AND VICTOR NISTOR

Introduction

239

(26)

1. Hochschild homology

240

(5)

2. Cyclic homology

245

(7)

3. Introducing topology

252

(2)

4. Algebras of pseudodifferential operators

254

(5)

5. Applications and examples

259

(1)

5.1. Manifolds with corners

259

(1)

5.2. Fibrations by manifolds with corners

260

(1)

5.3. Longitudinal symbols on foliations

260

(2)

References

262

(3)

Index and secondary index theory for flat bundles with duality

by ULRICH BUNKE AND XIAONAN MA

1. Introduction to the paper

265

(2)

2. The functor Lepsilon

267

(13)

2.1. Introduction and summary

267

(1)

2.2. Definition and first properties

268

(3)

2.3. Computation of Lepsilon(X)

271

(4)

2.4. The natural transformation to Kappa-theory

275

(1)

2.5. Push-forward for Lepsilon

276

(4)

3. The functor Lepsilon

280

(16)

3.1. Introduction and summary

280

(1)

3.2. Secondary Kappa-theory

281

(1)

3.3. The functor LR/Z

282

(1)

3.4. The functor L

283

(10)

3.5. Functorial properties

293

(3)

4. Eta homomorphisms

296

(9)

4.1. Introduction and summary

296

(1)

4.2. η-1R/Z:-1R/Z(M)-->R/Z

296

(1)

4.3. R-valued η-homomorphisms

297

(2)

4.4. Definition of the extended L-groups

299

(5)

4.5. Construction of η:Lepsilonex(M) -->R

304

(1)

5. The secondary index map

305

(17)

5.1. Introduction and summary

305

(1)

5.2. Adiabatic limits of eta invariants, the eta form

306

(8)

5.3. Definition of the secondary index maps

314

(2)

5.4. Functorial properties

316

(3)

5.5. The index map for L epsilon ex and L epsilon ex

319

(3)

6. Adiabatic limits - sketches of proofs

322

(18)

6.1. Remarks

322

(1)

6.2. The proof of Theorem 4.17

322

(8)

6.3. Proof of Theorem 5.9

330

(3)

6.4. Proof of Theorem 5.10

333

(2)

6.5. Proof of Theorem 5.11

335

(5)

References

340

(2)

Part II. Elliptic Boundary Value Problems

Toeplitz operators, and ellipticity of boundary value problems with global projection conditions

by B.-W. SCHULZE

Introduction

342

(6)

1. Elements of the classical calculus of boundary value problems

348

(37)

1.1. Pseudo-differential and Toeplitz operators on a closed manifold

348

(21)

1.2. Operators with the transmission property at the boundary

369

(14)

1.3. SL-elliptic boundary value problems

383

(2)

2. Ellipticity with global projection conditions

385

(16)

2.1. The index obstruction

385

(7)

2.2. A Toeplitz algebra of boundary value problems

392

(3)

2.3. Ellipticity, parametrices, and the Fredholm property

395

(5)

2.4. Reduction to the boundary

400

(1)

3. Transmission operators and Cauchy data spaces

401

(21)

3.1. Transmission operators

401

(7)

3.2. Examples

408

(2)

3.3. Spectral boundary value problems

410

(10)

3.4. Calderon-Seeley projections

420

(2)

4. Remarks on the edge calculus with global projection data

422

(4)

4.1. Boundary value problems without the transmission property

422

(1)

4.2. Edge problems

423

(3)

4.3. Analysis on manifolds with singularities

426

(1)

References

426

(4)

On the tangential oblique derivative problem - methods, results, open problems

by PETER POPIVANOV

Introduction

430

(2)

1. Subelliptic operators and their application to the tangential oblique derivative problem

432

(25)

1.1. Subelliptic estimates

432

(10)

1.2. Local solvability and microlocal analysis

442

(5)

1.3. Canonical transformations

447

(2)

1.4. The Calderon projector

449

(3)

1.5. Hölder regularity

452

(5)

2. Survey on the degenerate oblique derivative problem in Hölder, Sobolev, and Besov spaces

457

(12)

2.1. Tangential vector fields and solvability

458

(6)

2.2. Modified oblique derivative problem

464

(5)

3. Open problems

469

(1)

References

470

(2)

A note on boundary value problems on manifolds with cylindrical ends

by MARIUS MITREA AND VICTOR NISTOR

Introduction

472

(3)

1. Operators on manifolds with cylindrical ends

475

(3)

1.1. Manifolds with cylindrical ends and the Laplace operator

475

(1)

1.2. Operators that are translation invariant in a neighborhood of infinity

476

(2)

2. A spectrally invariant algebra

478

(4)

2.1. Operators that are almost translation invariant in a neighborhood of infinity

479

(2)

2.2. The "Δ + V trick"

481

(1)

3. Boundary layer potential integrals

482

(3)

3.1. Submanifolds with cylindrical ends

482

(1)

3.2. Boundary layer potential integrals

483

(2)

4. Layer potentials depending on a parameter

485

(4)

5. The Dirichlet problem

489

(3)

5.1. The Dirichlet-to-Neumann map

492

(1)

References

492

(3)

Relative elliptic theory

by VLADIMIR NAZAIKINSKII AND BORIS STERNIN

Introduction

495

(9)

Analytic Aspects of Relative Elliptic Theory

504

(1)

1. Classical objects. Lagrangian formalism

505

(14)

1.1. Lagrangian manifolds with measure

505

(4)

1.2. Composition and transposition formulas

509

(2)

1.3. Symbols

511

(5)

1.4. The product of symbols (twisted multiplication)

516

(3)

2. Quantization and the algebra of operator morphisms

519

(6)

2.1. The main operators

519

(3)

2.2. Composition formulas and adjoint operators

522

(1)

2.3. Algebras of operator morphisms

523

(2)

3. Elliptic morphisms and the index formula

525

(9)

3.1. The Green equation

526

(3)

3.2. Ellipticity and the Fredholm property

529

(4)

3.3. The index theorem for elliptic morphisms

533

(1)

4. The Sobolev problem

534

(5)

4.1. Sobolev problems and elliptic morphisms

534

(2)

4.2. The Fredholm property

536

(1)

4.3. The Sobolev problem and the adjoint problem

537

(1)

4.4. The index formula

538

(1)

Topological Aspects of Relative Elliptic Theory

539

(3)

5. Preliminaries

539

(1)

6. The metric trace of an elliptic operator

539

(3)

7. The pullback of an elliptic operator

542

(2)

7.1. The pullback of a pseudodifferential operator

542

(1)

7.2. The regularized pullback

543

(1)

7.3. The Riemann-Roch theorem for the regularized pullback

544

(1)

8. A finitely regularized pullback

544

(6)

8.1. A counterexample

544

(2)

8.2. The pullback for linear principal symbols

546

(2)

8.3. Pullbacks and exterior tensor products

548

(2)

9. The pullback of geometric operators

550

(5)

9.1. Group operators

550

(1)

9.2. The Euler and Hirzebruch operators

551

(3)

9.3. The Todd and Dirac operators

554

(1)

Appendix. Fourier Integral Operators

555

(4)

A.l. Homogeneous Lagrangian manifolds

555

(1)

A.2. Local description of homogeneous Lagrangian manifolds

556

(1)

A.3. Composition of homogeneous Lagrangian manifolds

556

(1)

A.4. Definition of Fourier integral operators

556

(2)

A.5. Pseudodifferential operators as Fourier integral operators

558

(1)

A.6. Boundedness theorems

558

(1)

A.7. Composition theorems

558

(1)

A.8. L2-adjoints of Fourier integral operators

559

(1)

A.9. Fourier integral operators on sections of vector bundles

559

(1)

References

559

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Aspects Of Boundary Problems In Analysis And Geometry

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