• Eigenfunctions

Eigenfunctions of the Laplacian on a Riemannian Manifold


Author: Steve Zelditch
Publisher: American Mathematical Soc.
ISBN: 1470410370
Category: Eigenfunctions
Page: 394
View: 7012
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.

    • Mathematics

The Laplacian on a Riemannian Manifold

An Introduction to Analysis on Manifolds
Author: Steven Rosenberg
Publisher: Cambridge University Press
ISBN: 9780521468312
Category: Mathematics
Page: 172
View: 4476
This text on analysis of Riemannian manifolds is aimed at students who have had a first course in differentiable manifolds.


    • Mathematics

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


Author: Urakawa Hajime
Publisher: World Scientific
ISBN: 9813109106
Category: Mathematics
Page: 312
View: 6018
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.

    • Mathematics

Eigenvalues in Riemannian Geometry


Author: Isaac Chavel
Publisher: Academic Press
ISBN: 9780080874340
Category: Mathematics
Page: 362
View: 6241
The basic goals of the book are: (i) to introduce the subject to those interested in discovering it, (ii) to coherently present a number of basic techniques and results, currently used in the subject, to those working in it, and (iii) to present some of the results that are attractive in their own right, and which lend themselves to a presentation not overburdened with technical machinery.

    • Mathematics

Riemannian Geometry

A Modern Introduction
Author: Isaac Chavel
Publisher: Cambridge University Press
ISBN: 1139452576
Category: Mathematics
Page: N.A
View: 4329
This book provides an introduction to Riemannian geometry, the geometry of curved spaces, for use in a graduate course. Requiring only an understanding of differentiable manifolds, the author covers the introductory ideas of Riemannian geometry followed by a selection of more specialized topics. Also featured are Notes and Exercises for each chapter, to develop and enrich the reader's appreciation of the subject. This second edition, first published in 2006, has a clearer treatment of many topics than the first edition, with new proofs of some theorems and a new chapter on the Riemannian geometry of surfaces. The main themes here are the effect of the curvature on the usual notions of classical Euclidean geometry, and the new notions and ideas motivated by curvature itself. Completely new themes created by curvature include the classical Rauch comparison theorem and its consequences in geometry and topology, and the interaction of microscopic behavior of the geometry with the macroscopic structure of the space.

    • Mathematics

Geometry of Manifolds


Author: N.A
Publisher: Academic Press
ISBN: 9780080873275
Category: Mathematics
Page: 272
View: 8399
Geometry of Manifolds

    • Mathematics

Heat Kernel and Analysis on Manifolds


Author: Alexander Grigoryan
Publisher: American Mathematical Soc.
ISBN: 0821849352
Category: Mathematics
Page: 482
View: 1642
"This volume contains the expanded lecture notes of courses taught at the Emile Borel Centre of the Henri Poincaré Institute (Paris). In the book, leading experts introduce recent research in their fields. The unifying theme is the study of heat kernels in various situations using related geometric and analytic tools. Topics include analysis of complex-coefficient elliptic operators, diffusions on fractals and on infinite-dimensional groups, heat kernel and isoperimetry on Riemannian manifolds, heat kernels and infinite dimensional analysis, diffusions and Sobolev-type spaces on metric spaces, quasi-regular mappings and p -Laplace operators, heat kernel and spherical inversion on SL 2 (C) , random walks and spectral geometry on crystal lattices, isoperimetric and isocapacitary inequalities, and generating function techniques for random walks on graphs."--Publisher's website.

    • Mathematics

Old and New Aspects in Spectral Geometry


Author: M.-E. Craioveanu,Mircea Puta,Themistocles Rassias
Publisher: Springer Science & Business Media
ISBN: 940172475X
Category: Mathematics
Page: 446
View: 4573
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.


    • Mathematics

Hangzhou Lectures on Eigenfunctions of the Laplacian (AM-188)


Author: Christopher D. Sogge
Publisher: Princeton University Press
ISBN: 1400850541
Category: Mathematics
Page: 208
View: 5238
Based on lectures given at Zhejiang University in Hangzhou, China, and Johns Hopkins University, this book introduces eigenfunctions on Riemannian manifolds. Christopher Sogge gives a proof of the sharp Weyl formula for the distribution of eigenvalues of Laplace-Beltrami operators, as well as an improved version of the Weyl formula, the Duistermaat-Guillemin theorem under natural assumptions on the geodesic flow. Sogge shows that there is quantum ergodicity of eigenfunctions if the geodesic flow is ergodic. Sogge begins with a treatment of the Hadamard parametrix before proving the first main result, the sharp Weyl formula. He avoids the use of Tauberian estimates and instead relies on sup-norm estimates for eigenfunctions. The author also gives a rapid introduction to the stationary phase and the basics of the theory of pseudodifferential operators and microlocal analysis. These are used to prove the Duistermaat-Guillemin theorem. Turning to the related topic of quantum ergodicity, Sogge demonstrates that if the long-term geodesic flow is uniformly distributed, most eigenfunctions exhibit a similar behavior, in the sense that their mass becomes equidistributed as their frequencies go to infinity.

    • Mathematics

A Panoramic View of Riemannian Geometry


Author: Marcel Berger
Publisher: Springer Science & Business Media
ISBN: 3642182453
Category: Mathematics
Page: 824
View: 2125
This book introduces readers to the living topics of Riemannian Geometry and details the main results known to date. The results are stated without detailed proofs but the main ideas involved are described, affording the reader a sweeping panoramic view of almost the entirety of the field. From the reviews "The book has intrinsic value for a student as well as for an experienced geometer. Additionally, it is really a compendium in Riemannian Geometry." --MATHEMATICAL REVIEWS

    • Mathematics

Dirac Operators in Riemannian Geometry


Author: Thomas Friedrich
Publisher: American Mathematical Soc.
ISBN: 0821820559
Category: Mathematics
Page: 195
View: 1656
Examines the Dirac operator on Riemannian manifolds, especially its connection with the underlying geometry and topology of the manifold. The presentation includes a review of Clifford algebras, spin groups and the spin representation, as well as a review of spin structures and spin [superscript C] structures. With this foundation established, the Dirac operator is defined and studied, with special attention to the cases of Hermitian manifolds and symmetric spaces. Then, certain analytic properties are established, including self-adjointness and the Fredholm property. An important link between the geometry and the analysis is provided by estimates for the eigenvalues of the Dirac operator in terms of the scalar curvature and the sectional curvature. Considerations of Killing spinors and solutions of the twistor equation on M lead to results about whether M is an Einstein manifold or conformally equivalent to one. Finally, in an appendix, Friedrich gives a concise introduction to the Seiberg-Witten invariants, which are a powerful tool for the study of four-manifolds. There is also an appendix reviewing principal bundles and connections.

    • Mathematics

Riemannian Geometry During the Second Half of the Twentieth Century


Author: Marcel Berger
Publisher: American Mathematical Soc.
ISBN: 0821820524
Category: Mathematics
Page: 182
View: 1747
In this book, Berger provides a survey of the main developments in Riemannian geometry in the last fifty years, focusing his main attention on the following five areas: Curvature and topology; the construction of and the classification of space forms; distinguished metrics, especially Einstein metrics; eigenvalues and eigenfunctions of the Laplacian; the study of periodic geodesics and the geodesic flow. Other topics are treated in less detail in a separate section.

    • Mathematics

The Atiyah-Patodi-Singer Index Theorem


Author: Richard Melrose
Publisher: CRC Press
ISBN: 1439864608
Category: Mathematics
Page: 392
View: 2615
Based on the lecture notes of a graduate course given at MIT, this sophisticated treatment leads to a variety of current research topics and will undoubtedly serve as a guide to further studies.

    • Mathematics

Spectral Theory and Geometry


Author: E. Brian Davies,Yu Safarov,International Centre for Mathematical Sciences
Publisher: Cambridge University Press
ISBN: 9780521777490
Category: Mathematics
Page: 328
View: 7734
Authoritative lectures from world experts on spectral theory and geometry.

    • Mathematics

Geometry and Spectra of Compact Riemann Surfaces


Author: Peter Buser
Publisher: Springer Science & Business Media
ISBN: 9780817649920
Category: Mathematics
Page: 456
View: 7915
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.

    • Mathematics

Spectral Geometry


Author: Alex Barnett
Publisher: American Mathematical Soc.
ISBN: 0821853198
Category: Mathematics
Page: 339
View: 4229
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.

    • Mathematics

Spectral Asymptotics in the Semi-Classical Limit


Author: M. Dimassi,J. Sjostrand
Publisher: Cambridge University Press
ISBN: 9780521665445
Category: Mathematics
Page: 227
View: 2319
This book presents the basic methods and applications in semiclassical approximation in the light of developments.

    • Mathematics

50 Years with Hardy Spaces

A Tribute to Victor Havin
Author: Anton Baranov,Sergei Kisliakov,Nikolai Nikolski
Publisher: Birkhäuser
ISBN: 3319590782
Category: Mathematics
Page: 484
View: 3962
Written in honor of Victor Havin (1933–2015), this volume presents a collection of surveys and original papers on harmonic and complex analysis, function spaces and related topics, authored by internationally recognized experts in the fields. It also features an illustrated scientific biography of Victor Havin, one of the leading analysts of the second half of the 20th century and founder of the Saint Petersburg Analysis Seminar. A complete list of his publications, as well as his public speech "Mathematics as a source of certainty and uncertainty", presented at the Doctor Honoris Causa ceremony at Linköping University, are also included.