Eigenvalues in Riemannian GeometryThe 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. |
Contents
| 1 | |
| 26 | |
Chapter III λ1 and Curvature | 55 |
Chapter IV Isoperimetric Inequalities | 85 |
Chapter V Eigenvalues and the Kinematic Measure | 113 |
Chapter VI The Heat Kernel for Compact Manifolds | 134 |
Chapter VII The Dirichlet Heat Kernel for Regular Domains | 158 |
Chapter VIII The Heat Kernel for Noncompact Manifolds | 179 |
Chapter IX Topological Perturbations with Negligible Effect | 207 |
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Common terms and phrases
argument assume boundary Cauchy–Schwarz inequality Cheeger compact closure compact Riemannian manifold compact support conjugacy class consider const constant sectional curvature continuous function convergence dA(w define denote diffeomorphic differential Dirichlet eigenvalue problem Dirichlet heat kernel domain Q easily eigenfunction equality estimate Euclidean exists finite fixed follows function f fundamental solution geodesic disk geometric given grad f Green's formula heat equation heat kernel hyperbolic hyperplane implies integral intersection isometry isoperimetric inequality Jacobi field Laplacian LEMMA Let Q lower bound lowest Dirichlet eigenvalue maximum principle n-dimensional Neumann eigenvalue nodal domain nontrivial normal obtain orthogonal positive constant PROOF OF THEOREM radius regular domain result Ricci curvature Riemannian manifold Riemannian metric satisfying Section I.5 sectional curvature sinh space variable sphere submanifolds tangent trace formula transformation uniformly upper bound valid vanishing vector field zero