Spectral Theory In Riemannian Geometry

Author : Olivier Lablée
Publisher : Erich Schmidt Verlag GmbH & Co. KG
ISBN 13 : 9783037191514
Total Pages : 204 pages
Book Rating : 4.11/5 ( download)

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Download or read book Spectral Theory in Riemannian Geometry written by Olivier Lablée and published by Erich Schmidt Verlag GmbH & Co. KG. This book was released on 2015 with total page 204 pages. Available in PDF, EPUB and Kindle. Book excerpt: Spectral theory is a diverse area of mathematics that derives its motivations, goals, and impetus from several sources. In particular, the spectral theory of the Laplacian on a compact Riemannian manifold is a central object in differential geometry. From a physical point a view, the Laplacian on a compact Riemannian manifold is a fundamental linear operator which describes numerous propagation phenomena: heat propagation, wave propagation, quantum dynamics, etc. Moreover, the spectrum of the Laplacian contains vast information about the geometry of the manifold. This book gives a self-contained introduction to spectral geometry on compact Riemannian manifolds. Starting with an overview of spectral theory on Hilbert spaces, the book proceeds to a description of the basic notions in Riemannian geometry. Then its makes its way to topics of main interests in spectral geometry. The topics presented include direct and inverse problems. Direct problems are concerned with computing or finding properties on the eigenvalues while the main issue in inverse problems is knowing the spectrum of the Laplacian, can we determine the geometry of the manifold? Addressed to students or young researchers, the present book is a first introduction to spectral theory applied to geometry. For readers interested in pursuing the subject further, this book will provide a basis for understanding principles, concepts, and developments of spectral geometry.

Author : E. Brian Davies
Publisher : Cambridge University Press
ISBN 13 : 0521777496
Total Pages : 344 pages
Book Rating : 4.90/5 ( download)

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Download or read book Spectral Theory and Geometry written by E. Brian Davies and published by Cambridge University Press. This book was released on 1999-09-30 with total page 344 pages. Available in PDF, EPUB and Kindle. Book excerpt: Authoritative lectures from world experts on spectral theory and geometry.

Author : Urakawa Hajime
Publisher : World Scientific
ISBN 13 : 9813109106
Total Pages : 312 pages
Book Rating : 4.00/5 ( download)

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Download or read book Spectral Geometry Of The Laplacian: Spectral Analysis And Differential Geometry Of The Laplacian written by Urakawa Hajime and published by World Scientific. This book was released on 2017-06-02 with total page 312 pages. Available in PDF, EPUB and Kindle. Book excerpt: The totality of the eigenvalues of the Laplacian of a compact Riemannian manifold is called the spectrum. We describe how the spectrum determines a Riemannian manifold. The continuity of the eigenvalue of the Laplacian, Cheeger and Yau's estimate of the first eigenvalue, the Lichnerowicz–Obata's theorem on the first eigenvalue, the Cheng's estimates of the kth eigenvalues, and Payne–Pólya–Weinberger's inequality of the Dirichlet eigenvalue of the Laplacian are also described. Then, the theorem of Colin de Verdière, that is, the spectrum determines the totality of all the lengths of closed geodesics is described. We give the V Guillemin and D Kazhdan's theorem which determines the Riemannian manifold of negative curvature.

Author : Pierre H. Berard
Publisher : Springer
ISBN 13 : 3540409580
Total Pages : 284 pages
Book Rating : 4.88/5 ( download)

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Download or read book Spectral Geometry written by Pierre H. Berard and published by Springer. This book was released on 2006-11-14 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Author : Steven Rosenberg
Publisher : Cambridge University Press
ISBN 13 : 9780521468312
Total Pages : 190 pages
Book Rating : 4.10/5 ( download)

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Download or read book The Laplacian on a Riemannian Manifold written by Steven Rosenberg and published by Cambridge University Press. This book was released on 1997-01-09 with total page 190 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

Author : Peter Buser
Publisher : Springer Science & Business Media
ISBN 13 : 0817649921
Total Pages : 473 pages
Book Rating : 4.20/5 ( download)

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Download or read book Geometry and Spectra of Compact Riemann Surfaces written by Peter Buser and published by Springer Science & Business Media. This book was released on 2010-10-29 with total page 473 pages. Available in PDF, EPUB and Kindle. Book excerpt: This monograph is a self-contained introduction to the geometry of Riemann Surfaces of constant curvature –1 and their length and eigenvalue spectra. It focuses on two subjects: the geometric theory of compact Riemann surfaces of genus greater than one, and the relationship of the Laplace operator with the geometry of such surfaces. Research workers and graduate students interested in compact Riemann surfaces will find here a number of useful tools and insights to apply to their investigations.

Author : David Borthwick
Publisher : Springer Nature
ISBN 13 : 3030380025
Total Pages : 339 pages
Book Rating : 4.21/5 ( download)

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Download or read book Spectral Theory written by David Borthwick and published by Springer Nature. This book was released on 2020-03-12 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: This textbook offers a concise introduction to spectral theory, designed for newcomers to functional analysis. Curating the content carefully, the author builds to a proof of the spectral theorem in the early part of the book. Subsequent chapters illustrate a variety of application areas, exploring key examples in detail. Readers looking to delve further into specialized topics will find ample references to classic and recent literature. Beginning with a brief introduction to functional analysis, the text focuses on unbounded operators and separable Hilbert spaces as the essential tools needed for the subsequent theory. A thorough discussion of the concepts of spectrum and resolvent follows, leading to a complete proof of the spectral theorem for unbounded self-adjoint operators. Applications of spectral theory to differential operators comprise the remaining four chapters. These chapters introduce the Dirichlet Laplacian operator, Schrödinger operators, operators on graphs, and the spectral theory of Riemannian manifolds. Spectral Theory offers a uniquely accessible introduction to ideas that invite further study in any number of different directions. A background in real and complex analysis is assumed; the author presents the requisite tools from functional analysis within the text. This introductory treatment would suit a functional analysis course intended as a pathway to linear PDE theory. Independent later chapters allow for flexibility in selecting applications to suit specific interests within a one-semester course.

Author : Werner Müller
Publisher : Springer
ISBN 13 : 3540477624
Total Pages : 169 pages
Book Rating : 4.24/5 ( download)

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Download or read book Manifolds with Cusps of Rank One written by Werner Müller and published by Springer. This book was released on 2006-11-15 with total page 169 pages. Available in PDF, EPUB and Kindle. Book excerpt: The manifolds investigated in this monograph are generalizations of (XX)-rank one locally symmetric spaces. In the first part of the book the author develops spectral theory for the differential Laplacian operator associated to the so-called generalized Dirac operators on manifolds with cusps of rank one. This includes the case of spinor Laplacians on (XX)-rank one locally symmetric spaces. The time-dependent approach to scattering theory is taken to derive the main results about the spectral resolution of these operators. The second part of the book deals with the derivation of an index formula for generalized Dirac operators on manifolds with cusps of rank one. This index formula is used to prove a conjecture of Hirzebruch concerning the relation of signature defects of cusps of Hilbert modular varieties and special values of L-series. This book is intended for readers working in the field of automorphic forms and analysis on non-compact Riemannian manifolds, and assumes a knowledge of PDE, scattering theory and harmonic analysis on semisimple Lie groups.

Author : Alexandre Girouard
Publisher : American Mathematical Soc.
ISBN 13 : 147042665X
Total Pages : 284 pages
Book Rating : 4.51/5 ( download)

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Download or read book Geometric and Computational Spectral Theory written by Alexandre Girouard and published by American Mathematical Soc.. This book was released on 2017-10-30 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: A co-publication of the AMS and Centre de Recherches Mathématiques The book is a collection of lecture notes and survey papers based on the mini-courses given by leading experts at the 2015 Séminaire de Mathématiques Supérieures on Geometric and Computational Spectral Theory, held from June 15–26, 2015, at the Centre de Recherches Mathématiques, Université de Montréal, Montréal, Quebec, Canada. The volume covers a broad variety of topics in spectral theory, highlighting its connections to differential geometry, mathematical physics and numerical analysis, bringing together the theoretical and computational approaches to spectral theory, and emphasizing the interplay between the two.

Author : M.-E. Craioveanu
Publisher : Springer Science & Business Media
ISBN 13 : 940172475X
Total Pages : 447 pages
Book Rating : 4.53/5 ( download)

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Download or read book Old and New Aspects in Spectral Geometry written by M.-E. Craioveanu and published by Springer Science & Business Media. This book was released on 2013-03-14 with total page 447 pages. Available in PDF, EPUB and Kindle. Book excerpt: It is known that to any Riemannian manifold (M, g ) , with or without boundary, one can associate certain fundamental objects. Among them are the Laplace-Beltrami opera tor and the Hodge-de Rham operators, which are natural [that is, they commute with the isometries of (M,g)], elliptic, self-adjoint second order differential operators acting on the space of real valued smooth functions on M and the spaces of smooth differential forms on M, respectively. If M is closed, the spectrum of each such operator is an infinite divergent sequence of real numbers, each eigenvalue being repeated according to its finite multiplicity. Spectral Geometry is concerned with the spectra of these operators, also the extent to which these spectra determine the geometry of (M, g) and the topology of M. This problem has been translated by several authors (most notably M. Kac). into the col loquial question "Can one hear the shape of a manifold?" because of its analogy with the wave equation. This terminology was inspired from earlier results of H. Weyl. It is known that the above spectra cannot completely determine either the geometry of (M , g) or the topology of M. For instance, there are examples of pairs of closed Riemannian manifolds with the same spectra corresponding to the Laplace-Beltrami operators, but which differ substantially in their geometry and which are even not homotopically equiva lent.