Elliptic boundary problems have enjoyed interest recently, espe cially among C• -algebraists and mathematical physicists who want to understand single aspects of the theory, such as the behaviour of Dirac operators and their solution spaces in the case of a non-trivial boundary. However, the theory of elliptic boundary problems by far has not achieved the same status as the theory of elliptic operators on closed (compact, without boundary) manifolds. The latter is nowadays rec ognized by many as a mathematical work of art and a very useful technical tool with applications to a multitude of mathematical con texts. Therefore, the theory of elliptic operators on closed manifolds is well-known not only to a small group of specialists in partial dif ferential equations, but also to a broad range of researchers who have specialized in other mathematical topics. Why is the theory of elliptic boundary problems, compared to that on closed manifolds, still lagging behind in popularity? Admittedly, from an analytical point of view, it is a jigsaw puzzle which has more pieces than does the elliptic theory on closed manifolds. But that is not the only reason. Front Cover 1 Title 4 Copyright 5 Contents 6 Preface 8 List of Notations 16 Part I. CLIFFORD ALGEBRAS AND Dirac OPERATORS 20 1. Clifford Algebras and Clifford Modules 22 2. Clifford Bundles and Compatible Connections 29 3. Dirac Operators 38 4. Dirac Laplacian and Connection Laplacian 45 5. Eudidean Examples 48 6. The Classical Dirac (Atiyah-Singer) Operators on Spin Manifolds 55 7. Dirac Operators and Chirality 59 8. Unique Continuation Property for Dirac Operators 62 9. Invertible Doubles 69 10. Clueing Constructions. Relative Index Theorem 78 PART II. ANALYTICAL AND TOPOLOGICAL TOOLS 84 11. Sobolev Spaces on Manifolds with Boundary 86 12. Calderon Projector for Dirac Operators 94 13. Existence of of Null Space Elements 114 14. Spectral Projections of Dirac Operators 124 15. Pseudo-Differential Grassmannians 130 16. The Homotopy Groups of the Space of Seif-Adjoint Fredhohn Operators 146 A. Elementary Decompositions and Deformations 146 B. The Homotopy Groups of C. 152 17. The Spectral Flow of Families of Seif-Adjoint Operators 157 A. Continuity of Eigenvalues 157 B. The Spectral Flow on Loops in F. 159 C. Spectral Flow and Index 164 D. Non-Vanishing Spectral Flow 176 Part III. APPLICATIONS 180 18. Elliptic Boundary Problems and Pseudo-Differential Projections 182 19. Regularity of Solutions of Elliptic Boundary Problems 199 20. Fredhoim Property of the Operator AR 207 21. Exchanges on the Boundary: Type Formulas and the Cobordism Theorem for Dirac Operators . 224 22. The Index Theorem for Atiyah-Patodi-Singer Problems 230 A. Preliminary Remarks 230 B. Heat Kernels on the Cylinder 233 C. Duhamel's principle. Heat Kernels on Manifolds with Boundary 250 D. Proof of the Index Formula 258 E. L2-Reformulation 261 F. The Odd-Dimensional Case. A Three-Dimensional Example 267 23. Some R2marks on the Index of Generalized Atiyah-Patodi-Singer Problems 272 24. Bojarski's Theorem. General Linear Conjugation Problems 281 25. Cutting and Pasting of Elliptic Operators 295 26. Dirac Operators on the Two-Sphere 301 Bibliography 308 Index 322 Back Cover 327 Clifford Algebras And Dirac Operations -- Analytical And Topological Tools -- Applications. Bernhelm Booss- Bavnbek, Krzysztof P. Wojciechowski. Includes Bibliographical References (p. [289]-302) And Index.
