Metric structures for Riemannian and non-Riemannian spaces

Mikhail Gromov, Jacques LaFontaine, Pierre Pansu, S. M. Bates, M. Katz, P. Pansu, S. Semmes9780817645823, 9780817645830, 0817645829, 0817645837

Metric theory has undergone a dramatic phase transition in the last decades when its focus moved from the foundations of real analysis to Riemannian geometry and algebraic topology, to the theory of infinite groups and probability theory.

The new wave began with seminal papers by Svarc and Milnor on the growth of groups and the spectacular proof of the rigidity of lattices by Mostow. This progress was followed by the creation of the asymptotic metric theory of infinite groups by Gromov.

The structural metric approach to the Riemannian category, tracing back to Cheeger’s thesis, pivots around the notion of the Gromov–Hausdorff distance between Riemannian manifolds. This distance organizes Riemannian manifolds of all possible topological types into a single connected moduli space, where convergence allows the collapse of dimension with unexpectedly rich geometry, as revealed in the work of Cheeger, Fukaya, Gromov and Perelman. Also, Gromov found metric structure within homotopy theory and thus introduced new invariants controlling combinatorial complexity of maps and spaces, such as the simplicial volume, which is responsible for degrees of maps between manifolds. During the same period, Banach spaces and probability theory underwent a geometric metamorphosis, stimulated by the Levy–Milman concentration phenomenon, encompassing the law of large numbers for metric spaces with measures and dimensions going to infinity.

The first stages of the new developments were presented in Gromov’s course in Paris, which turned into the famous “Green Book” by Lafontaine and Pansu (1979). The present English translation of that work has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices—by Gromov on Levy’s inequality, by Pansu on “quasiconvex” domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures—as well as an extensive bibliography and index round out this unique and beautiful book.

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Table of contents :
Series ……Page 1
Title page ……Page 2
Date-line ……Page 3
Titlepage of second printing, 2001 ……Page 4
Date-line of second printing, 2001 ……Page 5
Epigraph ……Page 6
Contents ……Page 8
Preface to the French Edition ……Page 12
Preface to the English Edition ……Page 14
Introduction: Metrics Everywhere ……Page 16
A. Length structures ……Page 22
B. Path metric spaces ……Page 27
C. Examples of path metric spaces ……Page 31
D. Arc-wise isometries ……Page 43
A. Topological review ……Page 48
B. Elementary properties of dilatations for spheres ……Page 51
C. Homotopy counting Lipschitz maps ……Page 56
D. Dilatation of sphere-valued mappings ……Page 62
E+ Degrees of short maps between compact and noncompact manifolds ……Page 76
A. Lipschitz and HausdorfF distance ……Page 92
B. The noncompact case ……Page 106
C. The Hausdorff-Lipschitz metric, quasi-isometries, and word metrics ……Page 110
D+ First-order metric invariants and ultralimits ……Page 115
E+ Convergence with control ……Page 119
A. A review of measures and mm spaces ……Page 134
B. $square_lambda$-convergence of mm spaces ……Page 137
C. Geometry of measures in metric spaces ……Page 145
D. Basic geometry of the space $mathcal{X}$ ……Page 150
E. Concentration phenomenon ……Page 161
F. Geometric invariants of measures related to concentration ……Page 202
G. Concentration, spectrum, and the spectral diameter ……Page 211
H. Observable distance $underline{H}_lambda$ on the space $mathcal{X}$ and concentration $X^n to X$ ……Page 221
I. The Lipschitz order on $mathcal{X}$, pyramids, and asymptotic concentration ……Page 233
J. Concentration versus dissipation ……Page 242
A. First, some history (in dimension 2) ……Page 260
B. Next, some questions in dimensions $geq 3$ ……Page 265
C. Norms on homology and Jacobi varieties ……Page 266
D. An application of geometric integration theory ……Page 282
E+ Unstable systolic inequalities and filling ……Page 285
F+ Finer inequalities and systoles of universal spaces ……Page 290
A. Precompactness ……Page 294
B. Growth of fundamental groups ……Page 300
C. The first Betti number ……Page 305
D. Small loops ……Page 309
E+ Applications of the packing inequalities ……Page 315
F+ On the nilpotency of $pi_1$ ……Page 316
G+ Simplicial volume and entropy ……Page 323
H+ Generalized simplicial norms and the metrization of homotopy theory ……Page 328
I+ Ricci curvature beyond coverings ……Page 337
A. Quasiregular mappings ……Page 342
B. Isoperimetric dimension of a manifold ……Page 343
C. Computations of isoperimetric dimension ……Page 348
D. Generalized quasiconformality ……Page 357
E+ The Varopoulos isoperimetric inequality ……Page 367
A. Application of Morse theory to loop spaces ……Page 372
B. Dilatation of mappings between simply connected manifolds ……Page 378
A. Invariant classes of metrics and the stability problem ……Page 386
B. Sign and the meaning of curvature ……Page 390
C. Elementary geometry of collapse ……Page 396
D. Convergence without collapse ……Page 405
E. Basic features of collapse ……Page 411
A “Quasiconvex” Domains in $mathbb{R}^n$ ……Page 414
B Metric Spaces and Mappings Seen at Many Scales ……Page 422
1. Euclidean spaces, hyperbolic spaces, and ideas from analysis ……Page 423
2. Quasimetrics, the doubling condition, and examples of metric spaces ……Page 425
3. Doubling measures and regular metric spaces, deformations of geometry, Riesz products and Riemann surfaces ……Page 432
4. Quasisymmetric mappings and deformations of geometry from doubling measures ……Page 438
5. Rest and recapitulation ……Page 443
6. Holder continuous functions on metric spaces ……Page 444
7. Metric spaces which are doubling ……Page 451
8. Spaces of homogeneous type ……Page 456
9. Holder continuity and mean oscillation ……Page 458
10. Vanishing mean oscillation ……Page 460
11. Bounded mean oscillation ……Page 464
12. Differentiability almost everywhere ……Page 466
14. Almost flat curves ……Page 469
15. Mappings that almost preserve distances ……Page 473
16. Almost flat hypersurfaces ……Page 476
17. The $A_infty$ condition for doubling measures ……Page 479
18. Quasisymmetric mappings and doubling measures ……Page 483
19. Metric doubling measures ……Page 485
20. Bi-Lipschitz embeddings ……Page 489
21. $A_1$ weights ……Page 491
24. Rectifiability ……Page 492
25. Uniform rectifiability ……Page 496
26. Stories from the past ……Page 498
27. Regular mappings ……Page 500
28. Big pieces of bi-Lipschitz mappings ……Page 501
29. Quantitative smoothness for Lipschitz functions ……Page 503
30. Smoothness of uniformly rectifiable sets ……Page 509
31. Comments about geometric complexity ……Page 511
32. The Maximal function ……Page 512
33. Covering lemmas ……Page 514
34. Lebesgue points ……Page 516
35. Differentiability almost everywhere ……Page 518
36. Finding Lipschitz pieces inside functions ……Page 523
38. Dyadic cubes ……Page 526
39. The Calderon-Zygmund approximation ……Page 528
40. The John-Nirenberg theorem ……Page 529
41. Reverse Holder inequalities ……Page 532
42. Two useful lemmas ……Page 534
43. Better methods for small oscillations ……Page 536
44. Real-variable methods and geometry ……Page 538
C Paul Levy’s Isoperimetric Inequality ……Page 540
D Systolically Free Manifolds ……Page 552
Bibliography ……Page 566
Glossary of Notation ……Page 596
Index ……Page 598
Cover ……Page 607