Old and New Aspects in Spectral Geometry by M.-E. CraioveanuOld and New Aspects in Spectral Geometry by M.-E. Craioveanu

Old and New Aspects in Spectral Geometry

byM.-E. Craioveanu, Mircea Puta, Themistocles RASSIAS

Paperback | December 6, 2010

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This work presents some classical as well as some very recent results and techniques concerning the spectral geometry corresponding to the Laplace-Beltrami operator and the Hodge-de Rham operators. It treats many topics that are not usually dealt with in this field, such as the continuous dependence of the eigenvalues with respect to the Riemannian metric in the CINFINITY-topology, and some of their consequences, such as Uhlenbeck's genericity theorem; examples of non-isometric flat tori in all dimensions greater than or equal to four; Gordon's classical technique for constructing isospectral closed Riemannian manifolds; a detailed presentation of Sunada's technique and Pesce's approach to isospectrality; Gordon and Webb's example of non-isometric convex domains in Rn (n>=4) that are isospectral for both Dirichlet and Neumann boundary conditions; the Chanillo-Trèves estimate for the first positive eigenvalue of the Hodge-de Rham operator, etc. Significant applications are developed, and many open problems, references and suggestions for further reading are given. Several themes for additional research are pointed out. Audience: This volume is designed as an introductory text for mathematicians and physicists interested in global analysis, analysis on manifolds, differential geometry, linear and multilinear algebra, and matrix theory. It is accessible to readers whose background includes basic Riemannian geometry and functional analysis. These mathematical prerequisites are covered in the first two chapters, thus making the book largely self-contained.
Title:Old and New Aspects in Spectral GeometryFormat:PaperbackDimensions:454 pages, 9.45 × 6.3 × 0 inPublished:December 6, 2010Publisher:Springer NetherlandsLanguage:English

The following ISBNs are associated with this title:

ISBN - 10:9048158370

ISBN - 13:9789048158379

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Table of Contents

Preface. 1. Introduction to Riemannian Manifolds. 2. Canonical Differential Operators Associated to a Riemannian Manifold. 3. Spectral Properties of the Laplace-Beltrami Operator and Applications. 4. Isospectral Closed Riemannian Manifolds. 5. Spectral Properties of the Laplacians for the De Rham Complex. 6. Applications to Geometry and Topology. 7. An Introduction to Witten-Helffer-Sjöstrand Theory. 8. Open Problems and Comments. Appendix. Subject Index.