UID:
almahu_9949984139202882
Format:
1 online resource (379 p.)
Edition:
2nd ed.
ISBN:
1-281-76328-4
,
9786611763282
,
0-08-087434-7
Series Statement:
Pure and applied mathematics ; 115
Content:
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.
Note:
Description based upon print version of record.
,
Front Cover; Eigenvalues in Riemannian Geometry; Copyright Page; Contents; Preface; Chapter I. The Laplacian; 1. Definitions and Preliminaries; 2. Green's Formulas; 3. Basic Facts for Eigenvalue Problems; 4. The Wave and Heat Equations; 5. Rayleigh and Max-Min Methods; Chapter II. The Basic Examples; 1. Some Generalities; 2. Tori; 3. Weyl's Formula for Bounded Domains in Rn; 4. Spheres and Real Projective Spaces; 5. Disks in Constant Curvature Space Forms; Chapter III. λ1 and Curvature; 1. Geodesics and Curvature; 2. Comparison Theorems for Sectional Curvature Bounded from Above
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3. Comparison Theorems for Ricci Curvature Bounded from Below4. Obata and Toponogov Theorems; Chapter IV. Isoperimetric Inequalities; 1. The Co-Area Formula; 2. The Faber-Krahn Inequality; 3. The Cheeger, Sobolev, and Isoperimetric Constants; 4. The Sobolev Constant and Eigenvalue, Eigenfunction, Estimates; 5 .The Constants and Estimates for the Closed Eigenvalue Problem; Chapter V. Eigenvalues and the Kinematic Measure; 1. The Analytic Inequality; 2. M. Berger's Isoembolic Inequality; 3. Cheeger and Isoperimetric Constants, and the Kinematic Measure
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Chapter VI. The Heat Kernel for Compact Manifolds1. Duhamel's Principle and Its Consequences; 2. The Heat Equation on R"; 3. The Minakshisundaram-Pleijel Recursion Formulas; 4. Existence of the Heat Kernel; 4. Upper Bounds for the Heat Kernel; Chapter VII. The Dirichlet Heat Kernel for Regular Domains; 1. Preliminaries; 2. The Dirichlet Heat Kernel for Regular Domains; 3. Duhamel's Principle; Chapter VIII. The Heat Kernel for Noncompact Manifolds; 1. The Maximum Principle, and Uniqueness Theorems, for the Heat Operator; 2. The Heat Kernel for Noncompact Manifolds
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3. Comparison Theorems for Heat KernelsChapter IX. Topological Perturbations with Negligible Effect; 1. Statement and Discussion of the Results; 2. Proof of Theorems 1-6; 3. Using Rayleigh's Characterization of Eigenvalues; 4. The Mathematics of Crushed Ice; Chapter X. Surfaces of Constant Negative Curvature; 1. Geometry of the Hyperbolic Plane; 2. The Heat Kernel of the Hyperbolic Plane; 3. Löbell Surfaces and the Estimates of P. Buser; 4. Low Eigenvalues and Short Geodesics; 4. Low Eigenvalues and Short Geodesics; 5. The Upper Half-Space Model of Hyperbolic Space
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Chapter XI. The Selberg Trace Formula1. Preliminaries; 2. The Pretrace Formula; 3. Applications of the Pretrace Formula; 5. Applications of the Trace Formula; 6. Concluding Remarks; Chapter XII. Miscellanea; 1. Volumes of Disks and Spheres; 2. The Fourier Transform; 3. The Poisson Summation Formula; 4. The Fourier Transform and the Heat Equation; 5. Eigenfunctions on Spheres and Hyperbolic Space; 6. Minimal Submanifolds of Euclidean Spaces and Spheres; 7. Normalization of Geometric Data; 8. Geodesic Coordinates; 9. The Levy-Gromov Isopenmetric Inequality
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10. Heat Conduction on the Euclidean Upper Half-Space
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English
Additional Edition:
ISBN 0-12-170640-0
Language:
English
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