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Quantitative bi-Lipschitz embeddings of bounded-curvature manifolds and orbifolds

Sylvester Eriksson-Bique

Geometry & Topology 22 (2018) 1961–2026
Bibliography
1 S Alexander, V Kapovitch, A Petrunin, Alexandrov geometry, in preparation
2 A Andoni, A Naor, O Neiman, Snowflake universality of Wasserstein spaces, preprint (2015) arXiv:1509.08677
3 P Assouad, Plongements lipschitziens dans Rn, Bull. Soc. Math. France 111 (1983) 429 MR763553
4 L Auslander, Bieberbach’s theorems on space groups and discrete uniform subgroups of Lie groups, Ann. of Math. 71 (1960) 579 MR0121423
5 M Bonk, U Lang, Bi-Lipschitz parameterization of surfaces, Math. Ann. 327 (2003) 135 MR2006006
6 J Bourgain, On Lipschitz embedding of finite metric spaces in Hilbert space, Israel J. Math. 52 (1985) 46 MR815600
7 D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) MR1835418
8 Y Burago, M Gromov, G Perel’man, A D Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) 3 MR1185284
9 P Buser, H Karcher, The Bieberbach case in Gromov’s almost flat manifold theorem, from: "Global differential geometry and global analysis" (editors D Ferus, W Kühnel, U Simon, B Wegner), Lecture Notes in Math. 838, Springer (1981) 82 MR636268
10 P Buser, H Karcher, Gromov’s almost flat manifolds, 81, Soc. Math. France (1981) 148 MR619537
11 J Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999) 428 MR1708448
12 J Cheeger, K Fukaya, M Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992) 327 MR1126118
13 M J Collins, On Jordan’s theorem for complex linear groups, J. Group Theory 10 (2007) 411 MR2334748
14 L J Corwin, F P Greenleaf, Representations of nilpotent Lie groups and their applications, I : Basic theory and examples, 18, Cambridge Univ. Press (1990) MR1070979
15 Y Ding, A restriction for singularities on collapsing orbifolds, ISRN Geom. (2011)
16 Y Ding, A restriction for singularities on collapsing orbifolds, preprint (2011) arXiv:1101.4444
17 K Fukaya, A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom. 28 (1988) 1 MR950552
18 K Fukaya, Collapsing Riemannian manifolds to ones with lower dimension, II, J. Math. Soc. Japan 41 (1989) 333 MR984756
19 K Fukaya, Hausdorff convergence of Riemannian manifolds and its applications, from: "Recent topics in differential and analytic geometry" (editor T Ochiai), Adv. Stud. Pure Math. 18, Academic Press (1990) 143 MR1145256
20 K Fukaya, Collapsing Riemannian manifolds and its applications, from: "Proceedings of the International Congress of Mathematicians, I" (editor I Satake), Math. Soc. Japan (1991) 491 MR1159236
21 P Ghanaat, M Min-Oo, E A Ruh, Local structure of Riemannian manifolds, Indiana Univ. Math. J. 39 (1990) 1305 MR1087193
22 R E Greene, H Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988) 119 MR917868
23 M Gromov, Almost flat manifolds, J. Differential Geom. 13 (1978) 231 MR540942
24 M Gromov, Metric structures for Riemannian and non-Riemannian spaces, 152, Birkhäuser (1999) MR1699320
25 K Grove, H Karcher, How to conjugate C1 –close group actions, Math. Z. 132 (1973) 11 MR0356104
26 I Haviv, O Regev, The Euclidean distortion of flat tori, J. Topol. Anal. 5 (2013) 205 MR3062945
27 J Heinonen, Lectures on analysis on metric spaces, Springer (2001) MR1800917
28 J Heinonen, Lectures on Lipschitz analysis, 100, Univ. of Jyväskylä (2005) MR2177410
29 J Heinonen, S Keith, Flat forms, bi-Lipschitz parameterizations, and smoothability of manifolds, Publ. Math. Inst. Hautes Études Sci. 113 (2011) 1 MR2805596
30 P Indyk, J Matoušek, Low-distortion embeddings of finite metric spaces, from: "Handbook of discrete and computational geometry" (editors J E Goodman, J O’Rourke), Chapman & Hall/CRC (2004) 177
31 V Kapovitch, Perelman’s stability theorem, from: "Surveys in differential geometry, XI" (editors J Cheeger, K Grove), International Press (2007) 103 MR2408265
32 H Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977) 509 MR0442975
33 S Khot, A Naor, Nonembeddability theorems via Fourier analysis, Math. Ann. 334 (2006) 821 MR2209259
34 B Kleiner, J Lott, Locally collapsed 3–manifolds, Astérisque 365, Soc. Math. France (2014) 7 MR3244329
35 T J Laakso, Ahlfors Q–regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality, Geom. Funct. Anal. 10 (2000) 111 MR1748917
36 U Lang, C Plaut, Bilipschitz embeddings of metric spaces into space forms, Geom. Dedicata 87 (2001) 285 MR1866853
37 U Lang, V Schroeder, Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal. 7 (1997) 535 MR1466337
38 K Luosto, Ultrametric spaces bi-Lipschitz embeddable in Rn, Fund. Math. 150 (1996) 25 MR1387955
39 J Luukkainen, H Movahedi-Lankarani, Minimal bi-Lipschitz embedding dimension of ultrametric spaces, Fund. Math. 144 (1994) 181 MR1273695
40 A I Mal’cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949) 9 MR0028842
41 J Matoušek, Bi-Lipschitz embeddings into low-dimensional Euclidean spaces, Comment. Math. Univ. Carolin. 31 (1990) 589 MR1078491
42 J Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976) 293 MR0425012
43 A Nagel, E M Stein, S Wainger, Balls and metrics defined by vector fields, I : Basic properties, Acta Math. 155 (1985) 103 MR793239
44 A Naor, An introduction to the Ribe program, Jpn. J. Math. 7 (2012) 167 MR2995229
45 A Naor, O Neiman, Assouad’s theorem with dimension independent of the snowflaking, Rev. Mat. Iberoam. 28 (2012) 1123 MR2990137
46 A Naor, Y Peres, O Schramm, S Sheffield, Markov chains in smooth Banach spaces and Gromov-hyperbolic metric spaces, Duke Math. J. 134 (2006) 165 MR2239346
47 I G Nikolaev, Bounded curvature closure of the set of compact Riemannian manifolds, Bull. Amer. Math. Soc. 24 (1991) 171 MR1056559
48 P Pansu, Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. 129 (1989) 1 MR979599
49 J G Ratcliffe, Foundations of hyperbolic manifolds, 149, Springer (2006) MR2249478
50 M Romney, Conformal Grushin spaces, Conform. Geom. Dyn. 20 (2016) 97 MR3492624
51 X Rong, On the fundamental groups of manifolds of positive sectional curvature, Ann. of Math. 143 (1996) 397 MR1381991
52 E A Ruh, Almost flat manifolds, J. Differential Geom. 17 (1982) 1 MR658470
53 S Semmes, Bi-Lipschitz mappings and strong A weights, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993) 211 MR1234732
54 S Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A–weights, Rev. Mat. Iberoamericana 12 (1996) 337 MR1402671
55 J Seo, A characterization of bi-Lipschitz embeddable metric spaces in terms of local bi-Lipschitz embeddability, Math. Res. Lett. 18 (2011) 1179 MR2915474
56 C Sormani, How Riemannian manifolds converge, from: "Metric and differential geometry" (editors X Dai, X Rong), Progr. Math. 297, Springer (2012) 91 MR3220440
57 T Tao, Hilbert’s fifth problem and related topics, 153, Amer. Math. Soc. (2014) MR3237440
58 W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979)
59 W P Thurston, Three-dimensional geometry and topology, I, 35, Princeton Univ. Press (1997) MR1435975
60 T Toro, Surfaces with generalized second fundamental form in L2 are Lipschitz manifolds, J. Differential Geom. 39 (1994) 65 MR1258915
61 T Toro, Geometric conditions and existence of bi-Lipschitz parameterizations, Duke Math. J. 77 (1995) 193 MR1317632
62 N T Varopoulos, L Saloff-Coste, T Coulhon, Analysis and geometry on groups, 100, Cambridge Univ. Press (1992) MR1218884