Spectral Geometry Of The Laplacian Spectral Analysis And Differential Geometry Of The Laplacian

Author : Urakawa Hajime
Publisher : World Scientific
Page : 312 pages
File Size : 54,6 Mb
Release : 2017-06-02
Category : Mathematics
ISBN : 9789813109100

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Spectral Geometry Of The Laplacian: Spectral Analysis And Differential Geometry Of The Laplacian by Urakawa Hajime Pdf

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
Page : 284 pages
File Size : 42,6 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540409588

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Spectral Geometry by Pierre H. Berard Pdf

Author : Michael Levitin,Dan Mangoubi,Iosif Polterovich
Publisher : American Mathematical Society
Page : 346 pages
File Size : 47,6 Mb
Release : 2023-11-30
Category : Mathematics
ISBN : 9781470475253

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Topics in Spectral Geometry by Michael Levitin,Dan Mangoubi,Iosif Polterovich Pdf

It is remarkable that various distinct physical phenomena, such as wave propagation, heat diffusion, electron movement in quantum mechanics, oscillations of fluid in a container, can be described using the same differential operator, the Laplacian. Spectral data (i.e., eigenvalues and eigenfunctions) of the Laplacian depend in a subtle way on the geometry of the underlying object, e.g., a Euclidean domain or a Riemannian manifold, on which the operator is defined. This dependence, or, rather, the interplay between the geometry and the spectrum, is the main subject of spectral geometry. Its roots can be traced to Ernst Chladni's experiments with vibrating plates, Lord Rayleigh's theory of sound, and Mark Kac's celebrated question “Can one hear the shape of a drum?” In the second half of the twentieth century spectral geometry emerged as a separate branch of geometric analysis. Nowadays it is a rapidly developing area of mathematics, with close connections to other fields, such as differential geometry, mathematical physics, partial differential equations, number theory, dynamical systems, and numerical analysis. This book can be used for a graduate or an advanced undergraduate course on spectral geometry, starting from the basics but at the same time covering some of the exciting recent developments which can be explained without too many prerequisites.

Author : Alex Barnett
Publisher : American Mathematical Soc.
Page : 354 pages
File Size : 53,8 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821853191

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Spectral Geometry by Alex Barnett Pdf

This volume contains the proceedings of the International Conference on Spectral Geometry, held July 19-23, 2010, at Dartmouth College, Dartmouth, New Hampshire. Eigenvalue problems involving the Laplace operator on manifolds have proven to be a consistently fertile area of geometric analysis with deep connections to number theory, physics, and applied mathematics. Key questions include the measures to which eigenfunctions of the Laplacian on a Riemannian manifold condense in the limit of large eigenvalue, and the extent to which the eigenvalues and eigenfunctions of a manifold encode its geometry. In this volume, research and expository articles, including those of the plenary speakers Peter Sarnak and Victor Guillemin, address the flurry of recent progress in such areas as quantum unique ergodicity, isospectrality, semiclassical measures, the geometry of nodal lines of eigenfunctions, methods of numerical computation, and spectra of quantum graphs. This volume also contains mini-courses on spectral theory for hyperbolic surfaces, semiclassical analysis, and orbifold spectral geometry that prepared the participants, especially graduate students and young researchers, for conference lectures.

Author : Steven Rosenberg
Publisher : Cambridge University Press
Page : 190 pages
File Size : 52,6 Mb
Release : 1997-01-09
Category : Mathematics
ISBN : 0521468310

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The Laplacian on a Riemannian Manifold by Steven Rosenberg Pdf

This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.

Author : AMS Symposium on the Geometry of the Laplace Operator$ (1979 : University of Hawaii at Manoa),American Mathematical Society
Publisher : American Mathematical Soc.
Page : 323 pages
File Size : 53,5 Mb
Release : 1980
Category : Mathematics
ISBN : 9780821814390

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Geometry of the Laplace Operator by AMS Symposium on the Geometry of the Laplace Operator$ (1979 : University of Hawaii at Manoa),American Mathematical Society Pdf

Author : Alexandre Girouard,Dmitry Jakobson,Michael Levitin,Nilima Nigam,Iosif Polterovich,Frédéric Rochon
Publisher : American Mathematical Soc.
Page : 284 pages
File Size : 49,6 Mb
Release : 2017-10-30
Category : Geometry, Differential
ISBN : 9781470426651

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Geometric and Computational Spectral Theory by Alexandre Girouard,Dmitry Jakobson,Michael Levitin,Nilima Nigam,Iosif Polterovich,Frédéric Rochon Pdf

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 : E. Brian Davies,Yu Safarov,London Mathematical Society,International Centre for Mathematical Sciences
Publisher : Cambridge University Press
Page : 344 pages
File Size : 45,6 Mb
Release : 1999-09-30
Category : Mathematics
ISBN : 9780521777490

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Spectral Theory and Geometry by E. Brian Davies,Yu Safarov,London Mathematical Society,International Centre for Mathematical Sciences Pdf

Authoritative lectures from world experts on spectral theory and geometry.

Author : Giuseppe Patanè
Publisher : Morgan & Claypool Publishers
Page : 141 pages
File Size : 46,7 Mb
Release : 2017-07-05
Category : Computers
ISBN : 9781681731407

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An Introduction to Laplacian Spectral Distances and Kernels by Giuseppe Patanè Pdf

In geometry processing and shape analysis, several applications have been addressed through the properties of the Laplacian spectral kernels and distances, such as commute time, biharmonic, diffusion, and wave distances. Within this context, this book is intended to provide a common background on the definition and computation of the Laplacian spectral kernels and distances for geometry processing and shape analysis. To this end, we define a unified representation of the isotropic and anisotropic discrete Laplacian operator on surfaces and volumes; then, we introduce the associated differential equations, i.e., the harmonic equation, the Laplacian eigenproblem, and the heat equation. Filtering the Laplacian spectrum, we introduce the Laplacian spectral distances, which generalize the commute-time, biharmonic, diffusion, and wave distances, and their discretization in terms of the Laplacian spectrum. As main applications, we discuss the design of smooth functions and the Laplacian smoothing of noisy scalar functions. All the reviewed numerical schemes are discussed and compared in terms of robustness, approximation accuracy, and computational cost, thus supporting the reader in the selection of the most appropriate with respect to shape representation, computational resources, and target application.

Author : Dirk Ferus,Ulrich Pinkall,Udo Simon,Berd Wegner
Publisher : Springer
Page : 289 pages
File Size : 49,5 Mb
Release : 2006-11-14
Category : Mathematics
ISBN : 9783540464457

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Global Differential Geometry and Global Analysis by Dirk Ferus,Ulrich Pinkall,Udo Simon,Berd Wegner Pdf

All papers appearing in this volume are original research articles and have not been published elsewhere. They meet the requirements that are necessary for publication in a good quality primary journal. E.Belchev, S.Hineva: On the minimal hypersurfaces of a locally symmetric manifold. -N.Blasic, N.Bokan, P.Gilkey: The spectral geometry of the Laplacian and the conformal Laplacian for manifolds with boundary. -J.Bolton, W.M.Oxbury, L.Vrancken, L.M. Woodward: Minimal immersions of RP2 into CPn. -W.Cieslak, A. Miernowski, W.Mozgawa: Isoptics of a strictly convex curve. -F.Dillen, L.Vrancken: Generalized Cayley surfaces. -A.Ferrandez, O.J.Garay, P.Lucas: On a certain class of conformally flat Euclidean hypersurfaces. -P.Gauduchon: Self-dual manifolds with non-negative Ricci operator. -B.Hajduk: On the obstruction group toexistence of Riemannian metrics of positive scalar curvature. -U.Hammenstaedt: Compact manifolds with 1/4-pinched negative curvature. -J.Jost, Xiaowei Peng: The geometry of moduli spaces of stable vector bundles over Riemannian surfaces. - O.Kowalski, F.Tricerri: A canonical connection for locally homogeneous Riemannian manifolds. -M.Kozlowski: Some improper affine spheres in A3. -R.Kusner: A maximum principle at infinity and the topology of complete embedded surfaces with constant mean curvature. -Anmin Li: Affine completeness and Euclidean completeness. -U.Lumiste: On submanifolds with parallel higher order fundamental form in Euclidean spaces. -A.Martinez, F.Milan: Convex affine surfaces with constant affine mean curvature. -M.Min-Oo, E.A.Ruh, P.Tondeur: Transversal curvature and tautness for Riemannian foliations. -S.Montiel, A.Ros: Schroedinger operators associated to a holomorphic map. -D.Motreanu: Generic existence of Morse functions on infinite dimensional Riemannian manifolds and applications. -B.Opozda: Some extensions of Radon's theorem.

Author : Olaf Post
Publisher : Springer
Page : 431 pages
File Size : 46,6 Mb
Release : 2012-01-05
Category : Mathematics
ISBN : 9783642238406

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Spectral Analysis on Graph-like Spaces by Olaf Post Pdf

Small-radius tubular structures have attracted considerable attention in the last few years, and are frequently used in different areas such as Mathematical Physics, Spectral Geometry and Global Analysis. In this monograph, we analyse Laplace-like operators on thin tubular structures ("graph-like spaces''), and their natural limits on metric graphs. In particular, we explore norm resolvent convergence, convergence of the spectra and resonances. Since the underlying spaces in the thin radius limit change, and become singular in the limit, we develop new tools such as norm convergence of operators acting in different Hilbert spaces, an extension of the concept of boundary triples to partial differential operators, and an abstract definition of resonances via boundary triples. These tools are formulated in an abstract framework, independent of the original problem of graph-like spaces, so that they can be applied in many other situations where the spaces are perturbed.

Author : Steve Zelditch
Publisher : American Mathematical Soc.
Page : 394 pages
File Size : 41,8 Mb
Release : 2017-12-12
Category : Eigenfunctions
ISBN : 9781470410377

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Eigenfunctions of the Laplacian on a Riemannian Manifold by Steve Zelditch Pdf

Eigenfunctions of the Laplacian of a Riemannian manifold can be described in terms of vibrating membranes as well as quantum energy eigenstates. This book is an introduction to both the local and global analysis of eigenfunctions. The local analysis of eigenfunctions pertains to the behavior of the eigenfunctions on wavelength scale balls. After re-scaling to a unit ball, the eigenfunctions resemble almost-harmonic functions. Global analysis refers to the use of wave equation methods to relate properties of eigenfunctions to properties of the geodesic flow. The emphasis is on the global methods and the use of Fourier integral operator methods to analyze norms and nodal sets of eigenfunctions. A somewhat unusual topic is the analytic continuation of eigenfunctions to Grauert tubes in the real analytic case, and the study of nodal sets in the complex domain. The book, which grew out of lectures given by the author at a CBMS conference in 2011, provides complete proofs of some model results, but more often it gives informal and intuitive explanations of proofs of fairly recent results. It conveys inter-related themes and results and offers an up-to-date comprehensive treatment of this important active area of research.

Author : M.-E. Craioveanu,Mircea Puta,Themistocles RASSIAS
Publisher : Springer Science & Business Media
Page : 447 pages
File Size : 48,9 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9789401724753

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Old and New Aspects in Spectral Geometry by M.-E. Craioveanu,Mircea Puta,Themistocles RASSIAS Pdf

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.

Author : Pierre Albin,Dmitry Jakobson, Frédéric Rochon
Publisher : American Mathematical Soc.
Page : 378 pages
File Size : 50,6 Mb
Release : 2014-12-01
Category : Mathematics
ISBN : 9781470410438

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Geometric and Spectral Analysis by Pierre Albin,Dmitry Jakobson, Frédéric Rochon Pdf

In 2012, the Centre de Recherches Mathématiques was at the center of many interesting developments in geometric and spectral analysis, with a thematic program on Geometric Analysis and Spectral Theory followed by a thematic year on Moduli Spaces, Extremality and Global Invariants. This volume contains original contributions as well as useful survey articles of recent developments by participants from three of the workshops organized during these programs: Geometry of Eigenvalues and Eigenfunctions, held from June 4-8, 2012; Manifolds of Metrics and Probabilistic Methods in Geometry and Analysis, held from July 2-6, 2012; and Spectral Invariants on Non-compact and Singular Spaces, held from July 23-27, 2012. The topics covered in this volume include Fourier integral operators, eigenfunctions, probability and analysis on singular spaces, complex geometry, Kähler-Einstein metrics, analytic torsion, and Strichartz estimates. This book is co-published with the Centre de Recherches Mathématiques.

Author : Olivier Lablée
Publisher : Erich Schmidt Verlag GmbH & Co. KG
Page : 204 pages
File Size : 42,7 Mb
Release : 2015
Category : Linear operators
ISBN : 3037191511

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Spectral Theory in Riemannian Geometry by Olivier Lablée Pdf

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.