Chavel I. — Eigenvalues in Riemannian geometry :: Электронная библиотека попечительского совета мехмата МГУ

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Chavel I. — Eigenvalues in Riemannian geometry

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Название: Eigenvalues in Riemannian geometry

Автор: Chavel I.

Аннотация:

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.

Язык:

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Издание: 2-nd edition

Год издания: 1984

Количество страниц: 379

Добавлена в каталог: 08.12.2013

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Предметный указатель

Associated Legendre equation and function      309
Berger isoembolic inequality      120
Berger — Kazdan inequality      114
Bochner — Lichnerowicz formula      83
Cheeger inequality, lowest Dirichlet eigenvalue      95
Cheeger inequality, lowest nonzero eigenvalue, closed problem      109
Cheeger inequality, lowest nonzero Neumann eigenvalue      259
Co-area formula      86
Comparison theorem, Bishop      68 71
Comparison theorem, Bonnet — Myers      73
Comparison theorem, Cheeger — Yau      194
Comparison theorem, Cheng      70 74
Comparison theorem, Debiard — Gaveau — Mazet      194
Comparison theorem, H. E. Rauch      67 74
Comparison theorem, Hadamard — Cartan      68
Comparison theorem, McKean      70
Comparison theorem, Morse — Schonberg      68
Comparison theorem, Obata      82
Comparison theorem, Toponogov      83
Conjugate point      62
Constant, Cheeger      95 109
Constant, isoperimetric      96 110
Constant, Sobolev      96 110
Convergence of heat flow to steady state, compact manifold      141
Convergence of heat flow to steady state, Dirichlet boundary data      170—172
Convergence of heat flow to steady state, Euclidean upper half-space      325—329
Convergence of heat flow to steady state, forms      339
Coordinates, Fermi, based on submanifold      319
Coordinates, Fermi, surface      247
Coordinates, geodesic spherical      65ff.
Coordinates, Riemann normal      317
Coordinates, sphere      37
Coordinates, spherical, Euclidean space      34
Coordinates, spherical, hyperbolic space      37
Cut locus      64
Cut point      64
Differential forms, Cheeger’s inequality and      342
Differential forms, deRham complex      334
Differential forms, eigenforms      339
Differential forms, eigenvalue estimates      342
Differential forms, Gauss — Bonnet theorem and heat equation      340—341
Differential forms, heat kernel      338
Differential forms, Hodge decomposition      335
Differential forms, Laplacian on      335
Dirichlet integral      14
Divergence theorem      6 7
Duhamel’s principle      137 145 165
Eigenfunction, $L^{2}- L^{\infty}$ estimates      102 112
Eigenvalues, 1-dimensional problems      10
Eigenvalues, closed problem      7
Eigenvalues, closed problem, bounds, lower      112 133 251 332
Eigenvalues, closed problem, bounds, upper      77 332
Eigenvalues, closed problem, bounds, upper, L $\ddot{o}$ bell surfaces      251
Eigenvalues, closed problem, calculation, projective space      36
Eigenvalues, closed problem, calculation, sphere      33
Eigenvalues, closed problem, calculation, torus      28
Eigenvalues, closed problem, lower, relation to short geodesics, Riemann surfaces      254—262
Eigenvalues, closed problem, lowest nonzero, bounds, lower      82 109 127 314 324 332
Eigenvalues, closed problem, lowest nonzero, bounds, upper, surface      94
Eigenvalues, closed problem, lowest nonzero, pinching theorem      92
Eigenvalues, closed problem, small, Riemann surface      247 300—302
Eigenvalues, Dirichlet problem      8
Eigenvalues, Dirichlet problem, bounds, lower      95 107 330
Eigenvalues, Dirichlet problem, calculation, disk, in space form      40—42
Eigenvalues, Dirichlet problem, calculation, domain, crystallographic group      33
Eigenvalues, Dirichlet problem, calculation, equilateral triangle      33
Eigenvalues, Dirichlet problem, lowest, bounds, lower      70 79 82 87 95 129
Eigenvalues, Dirichlet problem, lowest, bounds, lower, hyperbolic space      46
Eigenvalues, Dirichlet problem, lowest, bounds, upper      74 79 129
Eigenvalues, Dirichlet problem, lowest, bounds, upper, disk, hyperbolic space      80—82
Eigenvalues, Dirichlet problem, lowest, bounds, upper, L $\ddot{o}$ bell -piece      249
Eigenvalues, Dirichlet problem, lowest, disk, space form      42—45
Eigenvalues, Dirichlet problem, lowest, disk, space form, Euclidean space      45
Eigenvalues, Dirichlet problem, lowest, disk, space form, hyperbolic space      46
Eigenvalues, Dirichlet problem, lowest, disk, space form, sphere      50—54
Eigenvalues, domain monotonicity, vanishing Dirichlet data      17
Eigenvalues, domain monotonicity, vanishing Neumann data      18
Eigenvalues, lower bounds from upper bounds on heat kernel      157
Eigenvalues, max-min characterization      17
Eigenvalues, minimal immersions and      312—314
Eigenvalues, mixed eigenvalue problem      8
Eigenvalues, multiplicity of      24 35 104
Eigenvalues, Neumann problem      8
Eigenvalues, Neumann problem, calculation, disks in space form      40—42
Eigenvalues, Neumann problem, lowest nonzero, bounds, lower      251
Eigenvalues, Neumann problem, lowest nonzero, bounds, upper      94
Eigenvalues, Neumann problem, lowest nonzero, disks in space form      42—45
Eigenvalues, Neumann problem, lowest nonzero, disks in space form, Euclidean space      45
Eigenvalues, Neumann problem, lowest nonzero, disks in space form, hyperbolic space      46
Eigenvalues, Neumann problem, lowest nonzero, disks in space form, spheres      50
Eigenvalues, Rayleigh’s characterization      16
Eigenvalues, Sturm — Liouville decomposition, closed eigenvalue problem      139
Eigenvalues, Sturm — Liouville decomposition, Dirichlet eigenvalue problem      169
Eigenvalues, topological perturbations with negligible effect on      207—235
Fourier transform      304ff.
Fourier transform, Fourier inversion formula      305
Fourier transform, heat equation      307
Fourier transform, Plancherel formula      305
Fourier transform, Poisson summation formula      306
Fourier transform, Weyl asymptotic formula, torus      307
Function, admissible for eigenvalue problems      15
Function, directional derivative of      1
Function, gradient of      2
Function, Hessian of      82
Function, Laplacian of      3
Function, Sobolev space of      14
Function, weak derivative of      14
Fundamental solution, heat equation      see Heat kernel
Gamma function      303
Green’s formulas      6 7 165
Green’s function      173
Green’s function, Dirichlet problem      175
Green’s function, forms      339
Green’s function, relation to heat kernel      177
Heat equation, homogeneous      134
Heat equation, inhomogeneous      135
Heat equation, separation of variables      12
Heat kernel, almost Euclidean      159

Heat kernel, Brownian motion associated to      210ff.
Heat kernel, compact manifold, bounds, lower      333
Heat kernel, compact manifold, bounds, upper      112 133
Heat kernel, compact manifold, existence      151—154
Heat kernel, compact manifold, uniqueness      138
Heat kernel, comparison theorems      192—196
Heat kernel, Dirichlet      158
Heat kernel, Dirichlet, bounds, lower      195
Heat kernel, Dirichlet, bounds, upper      108 195 331—332
Heat kernel, Dirichlet, existence      164
Heat kernel, Dirichlet, uniqueness      167
Heat kernel, Euclidean space      142—148 307—308
Heat kernel, Fourier transform and      306—308
Heat kernel, noncompact manifolds      (see also Heat kernel Euclidean
Heat kernel, noncompact manifolds, bounds, lower      196
Heat kernel, noncompact manifolds, bounds, upper      180 196—206
Heat kernel, noncompact manifolds, Euclidean upper half-space      326
Heat kernel, noncompact manifolds, existence      188
Heat kernel, noncompact manifolds, space form      150 151
Heat kernel, noncompact manifolds, space form, hyperbolic plane      242—246
Heat kernel, topological perturbations with negligible effect on      207—235
Hyperbolic plane, geometry      239—241
Hyperbolic plane, heat kernel      242—246
Hyperbolic plane, upper half-plane model      262—264
Hyperbolic space, disk model      37
Hyperbolic space, geometry      266—271
Hyperbolic space, upper half-space model      264—265
Injectivity radius      118 120 247
Isoperimetric inequality, geometric      85 89 90 92 313
Isoperimetric inequality, geometric, L $\acute{e}$ vy — Gromov      91 322—325
Isoperimetric inequality, physical, closed eigenvalue problem, surfaces      94
Isoperimetric inequality, physical, Faber — Krahn      87
Isoperimetric inequality, physical, Faber — Krahn, minimal surfaces and      93
Isoperimetric inequality, physical, Faber — Krahn, variation of      91 93
Isoperimetric inequality, physical, Neumann eigenvalue problem      94
Jump relation      159 161
Kinematic measure      117
Kinematic measure, Liouville’s theorem      117
Kinematic measure, Santalo’s formula      125 130
Manifold, connection on      2
Manifold, connection on, Christoffel symbols of      4
Manifold, connection on, curvature, operator along geodesic      63
Manifold, connection on, curvature, tensor      59
Manifold, connection on, exponential map      57
Manifold, connection on, geodesic      57
Manifold, connection on, geodesic completeness      57
Manifold, Riemannian metric      2
Manifold, tangent bundle of      1
Manifold, tangent space to      1
Mathematics of crushed ice      233—238
Maximum principle, heat operator      139 166 169 180 182
Maximum principle, Laplace operator      20 127 329—330
Metaphysical principle      77
Minimal immersions and eigenvalues      312—314
Nodal domains      19
Nodal domains, Courant’s theorem      19 92
Nodal domains, Pleijel’s theorem      24 92
Nodal sets      19
Nodal sets, convexity of      24
Nodal sets, regularity of      23
Polya conjecture      33 330 333
Ricatti equation      72
Riemannian covering, deck transformation group      27
Riemannian manifold, curvature, Gauss      60
Riemannian manifold, curvature, Ricci      60
Riemannian manifold, curvature, scalar      60
Riemannian manifold, curvature, sectional      59
Riemannian manifold, distance between points of      39
Riemannian manifold, Gauss’s lemma      59
Riemannian manifold, L $\acute{e}$ vi — Civita connection on      2
Riemannian manifold, length of paths in      39
Riemannian manifold, open disk      40
Riemannian manifold, sphere in      40
Riemannian manifold, submanifold, cut point      320
Riemannian manifold, submanifold, focal point      320
Riemannian manifold, submanifold, mean curvature vector      310
Riemannian manifold, submanifold, normal bundle      224
Riemannian manifold, submanifold, second fundamental form      310
Riemannian manifold, submanifold, totally geodesic      131
Riemannian manifold, submanifold, Weingarten map      319
Riemannian manifold, surfaces, $theorema\quad egregium$       60
Riemannian manifold, surfaces, Gauss curvature      60
Riemannian manifold, surfaces, Gauss — Bonnet formula and theorem      60—61
Selberg trace formula      266—302
Selberg trace formula, pretrace formula      277
Selberg trace formula, pretrace formula, applications, electrostatics      282—288
Selberg trace formula, pretrace formula, applications, lattice-point estimates      279—281
Selberg trace formula, pretrace formula, applications, projections of dilating spheres      281—282
Selberg trace formula, trace formula      293
Selberg trace formula, trace formula, applications, Laplace and geodesic spectra      293 300
Selberg trace formula, trace formula, applications, small eigenvalues      300—302
Separation of variables, eigenfunctions and eigenvalues of disk in space form      40
Separation of variables, heat equation      12
Separation of variables, wave equation      11
surface      246
Surface, collar about geodesic      247
Surface, Riemann      246
Surface, Riemann, L $\ddot{o}$ bell      248—250
Vector field, along path, derivative of      55
Vector field, along path, parallel      56
Vector field, along path, parallel translation of      57
Vector field, covariant derivative of      2
Vector field, divergence of      3
Vector field, transverse      319
Visibility angle      125
Volumes, disks and spheres      303—304
Wave equation, finite propagation speed      198
Wave equation, separation of variables      11
Weyl’s asymptotic formula      9
Weyl’s asymptotic formula, 1-dimensional problems      10
Weyl’s asymptotic formula, bounded domain in $\mathbb{R}^{n}$       31
Weyl’s asymptotic formula, closed eigenvalue problem      155
Weyl’s asymptotic formula, Dirichlet eigenvalue problem      172
Weyl’s asymptotic formula, forms      339
Weyl’s asymptotic formula, Poisson summation formula and      306—307
Weyl’s asymptotic formula, torus      30
Wiedersehnsraum      123
Wiedersehnsraum, Blaschke conjecture      123
Wirtinger inequality      130

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