Table of contents for Metric structures for Riemannian and non-Riemannian spaces / Misha Gromov ; with appendices by M. Katz, P. Pansu, and S. Semmes ; English translation by Sean Michael Bates.


Bibliographic record and links to related information available from the Library of Congress catalog
Note: Electronic data is machine generated. May be incomplete or contain other coding.


Counter
Length Structures: Path Metric Spaces                           1
A.   Length structures .          . .
B,   Path metric spaces                                      6
C,. Examnples of path metric spaces ..   . . . . .  . .  ,  10
D.  Are-wise isometries .  ........ ...                    22
2 Degree and Dilatation                                        27
A. Topological review  .  .      ........            ...    27
B.   Elementary properties of dilatations for spheres. . . . .  .  30
C.   Homotopy counting Lipschitz maps .  . .  .     . . . .  35
D.   Dilatation of sphere-valued mappings .  . ..  . .  .   41
E+   Degrees of short maps between compact and
noncompact manifolds                                    55
3  Metric Structures on Families of Metric Spaces              71
A. Lipschitz and Hausdorff distance ..    ..  ..    ...     71
B.   The noncompactcase        .                .     . .  85
C.   The Hausdorff-Lipschitz metric,
quasi-isometries, and word metrics .      .. . .        89
De   First-order metric invariants
and  ultralimits  .. .... .                 ...    .    94
ES   Convergence with control  .     .......       ....     98
31   onvergence and Concentration of Metrics and Measures 113
A. A review of measures and mm spaces .                    1.  . . .  ..13
B.   OAconvergence of mm spaces                            116
C.   Geometry of measures in metric spaces ... .   .       124
D.   Basic geometry of the space X           . . . . . . .  129
E.   Concentration phenomenon . . . .  . . . . . . . . . . . . .  140
F.   Geometric invariants of measures
related to concentration              . ... 181
G. Concentration, spectrum, and
the spectral diameter                                 190
H.   Observable distance HA on the space X and concentration
XA n  X-i X. ...:                     . ........- II.. 200
.    The Lipschitz order on X, pyramids, and asymptotic con-
centration                                       . ...  212
J.   Concentration versus dissipation                      221
4  Loewner Rediscovered                                      239
A. First, some history (in dimension 2)    .               239
B.   Next, some questions in dimensions > 3         ..  .   . . 244
C.   Norms on homology and Jacobi varieties . . ....      245
D.   An application of geometric integration theory  .  . . . 261
"E" Unstable systolic inequalities and filling            26.  .
F+   Finer inequalities and systoles
of universal spaces                                   269
5 Manifolds with Bounded Ricci Curvature                     273
"A.  Precompactness                                 .   . 273
B.   Growth of fundamental groups . . . .  . . . . . . . . . . .  279
C.   The first Bettinumber .  . , . . .   . . .   . . . . . 284
D.   Small loops .. ..                                    288
E+   Applications of the packing inequalities .           294
F4   On the nilpotencyofr  ..   ..... .                 . . .  295
G4   Simplicial volume and entropy . . . ...      . ..     302
H+   Generalized simplicial norms and
the metrization of homotopy theory  ....... 307
I+   Ricci curvature beyond coverings     . .              316
6 Isoperinietric Inequalities and Amenability                321
A.   Quasiregular mappings...     . ...:           . .    321
B.   Isoperimetric dimension of a manifold           . .   322
C.   Computations of isoperimetric dimension  , . .327
D. Generalized quasiconformality                           3 . . .  ..  . . .  .  336
E+   The Varopoulos isoperimetric inequality .  . . .  . .  346
"7 Morse Theory and Minimal Models                           351
A.   Application of NMorse theory
to loop spaces                               ..       351
B.   Dilatation of mappings between
simply connected manifolds . . . . . . . . .          357
8+ Pinching and Collapse                                      365
A. Invariant classes of metrics
and the stability problem       . . .J. . . . .  . .   365
B    Sign and the meaning of curvature                      369
C.   Elenmntary geometry of collapse . . .                  375
Di Convergence without collapse                             384
E    Basic features of collapse .                           390
A  "Quasiconvex" Domains
inR                                                        393
B  Mjetric Spaces and Mappings Seen
at Many Scales                                             401
1.   Basic concepts and examples .       .   .       .   . 402
1. Euclidean spaces, hyperbolic spaces, and ideas from analysis 402
2   Quasinmetrics, the doubling condition, and
examples of metric spaces .  ...                       404
1 Doubling measures and regular metric spaces, deformations
of geometryi Riesz products and Riemann surfaces ..,   411
4. Quasisymmetric mappings and deformations of geometry from
do ubling measures                                     417
5. Rest and recaptuaton               .   .  .    . .     422
S    Analysis on general spaces                        .     23
6. HMder continuous functions on metric spaces .   .        23
7. Metric spaces which are doubling  ..,      ...       .  30
8. Spaces of homogeneous type ....                        435
9. 116older continuity and nmean oscillation  ..          437
10, Vanishing mean oscillation .... .                      439
11. Bounded mean oscillation .                              43
IIL Rigidity and structure ... . . .                        445
12. Differentiability almost everywhere                   . 445
13. Pause for reflection                                   448
14. Almost flat curves                                     448
15. Mappings that almost preserve distances . .    . .     452
16. Almost flat hypersurfaces                              45
17. The A, condition for doubling measures  .   ... .     458
18. Quasisymmnetric mappings and doubling measures  6. 42
19M NMtric doubling measures .   .    ..     .             464
20. Bi-Lipschitz embeddings ..     .                       468
21. A, weights                                        .       470
22, Interlude: bi-Lpschitz mappings between Cantor sets  .  471
23. Another moment of reflection  .471
24. Rectifiability                                         4 71
25  Uniform rectifiability                                475
26. Stories from the past .. :::.  .       , . .       . .  477
27   Regular mappings. .        .    ..       ,     , .    479
28   Big pieces of bi-Lipschitz mappings  ...     . . .    480
29. Quantitative smoothness for Lipschitz functions        482
30. Smoothness of uniformly rectifiable sets ,.            488
31. Comments about geometric complexity . . .  .,   . .  . 490
IV. An introduction to realvariable method                 491
32. The    aximal function              . . .              491
33   Covering emmas              .       .              .  493
34. Lebesgue points .              ......                  495
35   Differentiability almost everywhere . *.         ..   497
36. Finding Lipschitz pieces inside functions         ..   502
37, Maximal functions and snapshots  .                ,    505
38. Dyvadic cubes .  . ... .     .  . .  .                 505
39. The Calder6n Zygmund approximation .                   507
40. The John-Nirenberg theorem   .....      .    .     .   508
41   Reverse H6lder inequalities .      ...      . . . .   511l
42   Two useful lemmas                                     513
43. Better methods for small oscillations           . . .  515
44. Real-variable methods and geometry           . .     . 517
C  Paul Levy's Isoperimetric Inequality                       519
D  Systolical y Free Manifolds                                531



Library of Congress subject headings for this publication: Riemannian manifolds