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Chavel I. — Isoperimetric Inequalities : Differential Geometric and Analytic Perspectives
Chavel I. — Isoperimetric Inequalities : Differential Geometric and Analytic Perspectives



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Название: Isoperimetric Inequalities : Differential Geometric and Analytic Perspectives

Автор: Chavel I.

Аннотация:

This introduction treats the classical isoperimetric inequality in Euclidean space and contrasting rough inequalities in noncompact Riemannian manifolds. In Euclidean space the emphasis is on a most general form of the inequality sufficiently precise to characterize the case of equality, and in Riemannian manifolds the emphasis is on those qualitiative features of the inequality that provide insight into the coarse geometry at infinity of Riemannian manifolds. The treatment in Euclidean space features a number of proofs of the classical inequality in increasing generality, providing in the process a transition from the methods of classical differential geometry to those of modern geometric measure theory; and the treatment in Riemannian manifolds features discretization techniques, and applications to upper bounds of large time heat diffusion in Riemannian manifolds. The result is an introduction to the rich tapestry of ideas and techniques of isoperimetric inequalities, a subject that has its beginnings in classical antiquity and which continues to inspire fresh ideas in geometry and analysis to this very day — and beyond!


Язык: en

Рубрика: Математика/Геометрия и топология/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2001

Количество страниц: 268

Добавлена в каталог: 16.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$\delta$-cover      102
$\epsilon$-thickening      15 54
$\phi$-immeasurability      101
$\sigma$-algebra      100
Adams, C.      29
Adams, R.A.      29 224
Alexandres, A.D.      51
Almgren, F.      51 99
Approximate identity      229
Arc length function      4
Area, integral geometric      53
Area, lower      125
Area, Minkowski      58
Area, of boundaries in graphs      142
Area, of metric sphere in Riemannian manifold      125
Area, surface      2 3 17 25 31
Arede, T.      224
Aubin, T.      184 224
Azencott, R.      224
Baemstein, A.      99
Bakry, D.      183 254
Bandle, C.      183 254
Barlow, M.      254
Bendabdallah, A.      224
Benjamini, I.      29 233
Berard, P.H.      184 254
Berger, M.      29 99
Besson, C.      254
Blaschke, W.      28
Boundaries in graphs      142
Boundaries in graphs, area of      142
Boundary conditions, vanishing Dirichlet      196
Bounded geometry, for graphs      140
Bounded geometry, for Riemannian manifolds      136
Brooks, R.      232
Burago, Y.      3 28 99
Burchard, A.      99
Buser, P.      155
Cao, J.      29
Caratheodory, C.      99
Carleman, T.      5I
Carlen, E.A.      254
Carron, G.      184
Chavel, I.      29 99 155 183—185 224 233 253 254
Cheeger's constant      127
Cheeger's constant, in compact case      162
Cheeger's constant, modified      130
Cheeger, J.      183 253
Cheng, S.Y.      183
Circumdisk      57
Circumradius      53
Combinatorial, length of      140
Combinatorial, metric      140
Combinatorial, path      140
Complete, geodesically      120
Connection      121
Connection, Levi-Civita      121
Constant, alternate Sobolev, in compact case      163
Constant, Cheeger's      127
Constant, Cheeger's, in compact case      162
Constant, Cheeger's, modified      130
Constant, isoperimetric, modified, of Riemannian manifold      130
Constant, isoperimetric, of $\mathbb R^n$      44 45
Constant, isoperimetric, of compact Riemannian manifold      162
Constant, isoperimetric, of graphs      144
Constant, isoperimetric, of Riemannian manifold      127
Constant, Lipschitz      20
Constant, Sobolev, in compact case      162
Constant, Sobolev, of $\mathbb R^n$      44 45
Constant, Sobolev, of graphs      173
Constant, Sobolev, of Riemannian manifold      131
Convergence of sets in $L^1$      81
Convex, function      22
Convex, hull of      28
Convex, linear combination      27
Convex, set      27
Coordinates, Euclidean      17
Coordinates, geodesic spherical      125
Coordinates, Riemann normal      123
Coulhon, T.      155 183 184 233 254
Countable additivity      100
Countably subadditive      101
Courant, R.      28
Croke, C.      155
Curvature, constant      123
Curvature, Gauss      32
Curvature, mean      32 36 44
Curvature, of path in $\mathbb R^2$      5
Curvature, Ricci      122
Curvature, sectional      122
Curvature, tensor      121
Cut locus, domain inside, of a point      125
Davies, E.B.      224 254
Dierkes, U.      28
Differential operator, elliptic      19
Differential operator, uniformly      19
Dirichlet energy integral      195
Discretization of Riemannian manifold      147
Discretization of Riemannian manifold, covering radius of      147
Discretization of Riemannian manifold, separation radius of      147
Distance, combinatorial, between points      140
Distance, Hausdorff, between compact sets      54
Distance, Riemannian, between points      119
Dodziuk, J.      184 224 254
Domain      3
Domain, Lipschitz      87
Exponential map      120
Faber, C.      99
Fabes, E.      254
Federer, H.      29 51 99 116 183
Fegan, H.D.      224
Feldman, E.A.      184 233 254
Figiel, T.      29
Fischer, H.R.      224
Fleming, W.H.      51 183
Formula, Cauchy's, for area of boundaries of convex domains      26
Formula, co-area for perimeter      89
Formula, co-area for smooth functions      25
Formula, co-area for smooth mappings      25
Fourier transform      203
Frenet formulae      5
Function, characteristic      13
Function, differential of, on graphs      144 170
Function, discretization of      178
Function, gradient of, on Riemannian manifold      157
Function, indicator      13
Function, Laplacian of, acting as $L^2$ distribution      198
Function, Laplacian of, Friedrichs extension of      195
Function, Laplacian of, on Riemannian manifold      193
Function, Lipschitz      19
Function, mean value      162
Function, of bounded variation      82
Function, polynomial decay      252
Function, smoothing of      179
Function, weak derivative of      23
Function, weak derivative of, $L^2$      195
Functional, isoperimetric      36
Fundamental form, first      31
Fundamental form, second      32
Fundamental tone      78
Gaffney, M.P.      224
Gage, M.      29
Gallot, S.      254
Gauss map      32
Geodesic      119
Geodesic, completeness      120
Geodesic, flow      126
Geodesic, flow, kiuematic density of      127
Geodesic, flow, Liouville's theorem      127
Geodesic, spherical coordinates      125
Gilbarg, D.      29 224
Giusti, E.      99 100
Graph      140
Graph, Cayley, of group      141
Graph, combinatorial path in      140
Graph, distance between points      140
Graph, with bounded geometry      140
Grigor'yan, A.A.      155 156 184 224 233 254
Gromov, M.      29 51 184 253
Hadwiger, H.      28
Hasminkii, R.Z.      224
Hausdorff dimension      105
Hausdorff distance between compact sets      54
Hausdorff measure      102
Hausdorff measure, standard      104
Heat equation, homogeneous      201
Heat equation, inhomogeneous      201
Heat equation, uniqueness property      212
Heat flow      214
Heat kernel      202
Heat kernel, closed      208
Heat kernel, conservation of heat property of      208
Heat kernel, Dirichlet      209
Heat kernel, Feller property of      208
Heat kernel, minimal positive      211
Heat kernel, positive      208
Heat kernel, stochastic completeness of      208
Heat kernel, symmetric      208
Heat operator associated with the Laplacian      201
Hildebrandt, S.      28
Hirsch, M.W.      29
Holopainen, I.      156
Hoof, H.      51 155
Howards, H.      29
Hurwitz, A.      29
Hutchings, M.      29
Hyperplane, separation by      28
Hyperplane, strict      28
Hyperplane, supporting      28
Ichihara, K.      224
Incompressible flow      39
Inequality, arithmetic-geometric mean      38 49
Inequality, Bonnesen      4
Inequality, Brunn — Minkowski      62
Inequality, Buser's      136
Inequality, Crake's      136
Inequality, eigenvalue $\Lambda$-isoperimetric      169
Inequality, Faber — Krahn      79
Inequality, Faber — Krahn, of Euclidean type      166
Inequality, geometric g-isoperimetric      168
Inequality, isodiametric      61
Inequality, isoperimetric      2—4
Inequality, isoperimetric, eigenvalue $\Lambda$-      249
Inequality, isoperimetric, for $C^1$ surface area      47 51
Inequality, isoperimetric, for Minkowski area      63
Inequality, isoperimetric, for Minkowski area in $\mathbb R^n$      77
Inequality, isoperimetric, for Minkowski length in $\mathbb R^2$      74
Inequality, isoperimetric, for perimeter      94
Inequality, isoperimetric, geometric g-      249
Inequality, isoperimetric, in $\mathbb R^2$      8 9
Inequality, isoperimetric, in model space      4
Inequality, Wirtinger's      8 29
Injectivity radius of point      120
Injectivity radius of Riemannian manifold      120
Inradius      130
Isodiametric inequality      61
Isometry, of Riemannian manifolds      123
Isometry, quasi-, of metric spaces      140
Isometry, quasi-, of Riemannian manifolds      128
Isometry, rough, of metric spaces      142
Isoperimetric constant of $\mathbb R^n$      44 45
Isoperimetric constant of compact Riemannian manifold      162
Isoperimetric constant of graphs      144
Isoperimetric constant of Riemannian manifold      127
Isoperimetric constant, modified, of Riemannian manifold      130
Isoperimetric function      254
Isoperimetric functional      1 11 36 117
Isoperimetric functional, $C^k$ external of the      31
Isoperimetric functional, uniqueness of $C^2$ extremal of      39
Isoperimetric inequality, eigenvalue $\Lambda$-      249
Isoperimetric inequality, for $C^1$ surface area      47 51
Isoperimetric inequality, for Minkowski area      63
Isoperimetric inequality, for Minkowski area in $\mathbb R^n$      77
Isoperimetric inequality, for Minkowski length in $\mathbb R^2$      74
Isoperimetric inequality, for perimeter      94
Isoperimetric inequality, geometric g-      249
Isoperimetric inequality, in $\mathbb R^2$      8 9
Isoperimetric inequality, in model space      4
Isoperimetric inequality, problem      2
Isoperimetric inequality, problem, in model space      4 29
Isoperimetric inequality, problem, solution to the $C^k$      31
Isoperimetric inequality, problem, uniqueness of $C^2$ solution to      36
Jungster, J.J.      224
Kanai, M.      154 156 177 184
Karp, L.      224 253
Kawohl, B.      99 183
Knothe, H.      51
Krahn, E.      99
Kuerster, A.      28
Kusuoka, S.      254
Laplacian on Riemannian manifold      193
Laplacian on Riemannian manifold, acting as $L^2$ distribution      198
Laplacian on Riemannian manifold, Dirichlet      196
Laplacian on Riemannian manifold, Friedrichs extension of      195
Lebesgue set      96
Ledoux, M.      183
Length of path in Riemannian manifold      119
Length of vector in Riemannian metric      118
Li, P.      183 222 224 253 254
Lieb, E.H.      29 99
Liebmann, H.      31
Lindenstrauss, J.      29
Linear operator, adjoint of      185
Linear operator, closable      185
Linear operator, closed      185
Linear operator, closure of      185
Linear operator, extension of      185
Linear operator, extension of, minimal      185
Linear operator, resolvent of      187
Linear operator, resolvent set of      187
Linear operator, self-adjoint      186
Linear operator, self-adjoint, approximate eigenfuctions of      189
Linear operator, self-adjoint, essentially      186
Linear operator, self-adjoint, spectral family of      187
Linear operator, semibounded from below      186
Linear operator, semibounded from below, Friedrichs extension of      186
Linear operator, spectrum of      187
Linear operator, spectrum of, discrete      188
Linear operator, spectrum of, essential      188
Linear operator, spectrum of, point      188
Linear operator, symmetric      186
Linear operator, symmetric, nonnegative      186
Lipschitz constant      20
Lipschitz domain      87
Lipschitz function      19
Loss, M.      29 99
Mandouvalos, N.      224
Manifold, closed      161
Manifold, compact      161
Maz'ya, V.G.      51
Measure      100
Measure, $R^k$-valued      15
Measure, $R^k$-valued, absolutely continuous      15
Measure, $R^k$-valued, Radon      16
Measure, $R^k$-valued, total variation of      15
Measure, counting      142
Measure, Hausdorff      102
Measure, Hausdorff, $\delta$-      102
Measure, Hausdorff, standard      104
Measure, Hausdorff, standard $\delta$-      104
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