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

Eriksson-Bique, Sylvester¹

¹ Department of Mathematics, UCLA, Los Angeles, CA, United States

Geometry & topology, Tome 22 (2018) no. 4, pp. 1961-2026

Résumé

We construct bi-Lipschitz embeddings into Euclidean space for bounded-diameter subsets of manifolds and orbifolds of bounded curvature. The distortion and dimension of such embeddings is bounded by diameter, curvature and dimension alone. We also construct global bi-Lipschitz embeddings for spaces of the form ℝn∕Γ, where Γ is a discrete group acting properly discontinuously and by isometries on ℝn. This generalizes results of Naor and Khot. Our approach is based on analyzing the structure of a bounded-curvature manifold at various scales by specializing methods from collapsing theory to a certain class of model spaces. In the process, we develop tools to prove collapsing theory results using algebraic techniques.

Détail

DOI : 10.2140/gt.2018.22.1961

Classification : 30L05, 51F99, 53C21, 20H15, 53B20
Keywords: bilipschitz, sectional curvature, Alexandrov, collapsing theory

Bibliographie

[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)

[3] P Assouad, Plongements lipschitziens dans Rn, Bull. Soc. Math. France 111 (1983) 429

[4] L Auslander, Bieberbach’s theorems on space groups and discrete uniform subgroups of Lie groups, Ann. of Math. 71 (1960) 579 | DOI

[5] M Bonk, U Lang, Bi-Lipschitz parameterization of surfaces, Math. Ann. 327 (2003) 135 | DOI

[6] J Bourgain, On Lipschitz embedding of finite metric spaces in Hilbert space, Israel J. Math. 52 (1985) 46 | DOI

[7] D Burago, Y Burago, S Ivanov, A course in metric geometry, 33, Amer. Math. Soc. (2001) | DOI

[8] Y Burago, M Gromov, G Perel’Man, A D Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) 3

[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 | DOI

[10] P Buser, H Karcher, Gromov’s almost flat manifolds, 81, Soc. Math. France (1981) 148

[11] J Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999) 428 | DOI

[12] J Cheeger, K Fukaya, M Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992) 327 | DOI

[13] M J Collins, On Jordan’s theorem for complex linear groups, J. Group Theory 10 (2007) 411 | DOI

[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)

[15] Y Ding, A restriction for singularities on collapsing orbifolds, ISRN Geom. (2011) | DOI

[16] Y Ding, A restriction for singularities on collapsing orbifolds, preprint (2011)

[17] K Fukaya, A boundary of the set of the Riemannian manifolds with bounded curvatures and diameters, J. Differential Geom. 28 (1988) 1

[18] K Fukaya, Collapsing Riemannian manifolds to ones with lower dimension, II, J. Math. Soc. Japan 41 (1989) 333 | DOI

[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 | DOI

[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

[21] P Ghanaat, M Min-Oo, E A Ruh, Local structure of Riemannian manifolds, Indiana Univ. Math. J. 39 (1990) 1305 | DOI

[22] R E Greene, H Wu, Lipschitz convergence of Riemannian manifolds, Pacific J. Math. 131 (1988) 119 | DOI

[23] M Gromov, Almost flat manifolds, J. Differential Geom. 13 (1978) 231

[24] M Gromov, Metric structures for Riemannian and non-Riemannian spaces, 152, Birkhäuser (1999) | DOI

[25] K Grove, H Karcher, How to conjugate C1 –close group actions, Math. Z. 132 (1973) 11 | DOI

[26] I Haviv, O Regev, The Euclidean distortion of flat tori, J. Topol. Anal. 5 (2013) 205 | DOI

[27] J Heinonen, Lectures on analysis on metric spaces, Springer (2001) | DOI

[28] J Heinonen, Lectures on Lipschitz analysis, 100, Univ. of Jyväskylä (2005)

[29] J Heinonen, S Keith, Flat forms, bi-Lipschitz parameterizations, and smoothability of manifolds, Publ. Math. Inst. Hautes Études Sci. 113 (2011) 1 | DOI

[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 | DOI

[32] H Karcher, Riemannian center of mass and mollifier smoothing, Comm. Pure Appl. Math. 30 (1977) 509 | DOI

[33] S Khot, A Naor, Nonembeddability theorems via Fourier analysis, Math. Ann. 334 (2006) 821 | DOI

[34] B Kleiner, J Lott, Locally collapsed 3–manifolds, Astérisque 365, Soc. Math. France (2014) 7

[35] T J Laakso, Ahlfors Q–regular spaces with arbitrary Q > 1 admitting weak Poincaré inequality, Geom. Funct. Anal. 10 (2000) 111 | DOI

[36] U Lang, C Plaut, Bilipschitz embeddings of metric spaces into space forms, Geom. Dedicata 87 (2001) 285 | DOI

[37] U Lang, V Schroeder, Kirszbraun’s theorem and metric spaces of bounded curvature, Geom. Funct. Anal. 7 (1997) 535 | DOI

[38] K Luosto, Ultrametric spaces bi-Lipschitz embeddable in Rn, Fund. Math. 150 (1996) 25

[39] J Luukkainen, H Movahedi-Lankarani, Minimal bi-Lipschitz embedding dimension of ultrametric spaces, Fund. Math. 144 (1994) 181

[40] A I Mal’Cev, On a class of homogeneous spaces, Izvestiya Akad. Nauk. SSSR. Ser. Mat. 13 (1949) 9

[41] J Matoušek, Bi-Lipschitz embeddings into low-dimensional Euclidean spaces, Comment. Math. Univ. Carolin. 31 (1990) 589

[42] J Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976) 293 | DOI

[43] A Nagel, E M Stein, S Wainger, Balls and metrics defined by vector fields, I : Basic properties, Acta Math. 155 (1985) 103 | DOI

[44] A Naor, An introduction to the Ribe program, Jpn. J. Math. 7 (2012) 167 | DOI

[45] A Naor, O Neiman, Assouad’s theorem with dimension independent of the snowflaking, Rev. Mat. Iberoam. 28 (2012) 1123 | DOI

[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 | DOI

[47] I G Nikolaev, Bounded curvature closure of the set of compact Riemannian manifolds, Bull. Amer. Math. Soc. 24 (1991) 171 | DOI

[48] P Pansu, Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. of Math. 129 (1989) 1 | DOI

[49] J G Ratcliffe, Foundations of hyperbolic manifolds, 149, Springer (2006) | DOI

[50] M Romney, Conformal Grushin spaces, Conform. Geom. Dyn. 20 (2016) 97 | DOI

[51] X Rong, On the fundamental groups of manifolds of positive sectional curvature, Ann. of Math. 143 (1996) 397 | DOI

[52] E A Ruh, Almost flat manifolds, J. Differential Geom. 17 (1982) 1

[53] S Semmes, Bi-Lipschitz mappings and strong A∞ weights, Ann. Acad. Sci. Fenn. Ser. A I Math. 18 (1993) 211

[54] S Semmes, On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A∞–weights, Rev. Mat. Iberoamericana 12 (1996) 337 | DOI

[55] J Seo, A characterization of bi-Lipschitz embeddable metric spaces in terms of local bi-Lipschitz embeddability, Math. Res. Lett. 18 (2011) 1179 | DOI

[56] C Sormani, How Riemannian manifolds converge, from: "Metric and differential geometry" (editors X Dai, X Rong), Progr. Math. 297, Springer (2012) 91 | DOI

[57] T Tao, Hilbert’s fifth problem and related topics, 153, Amer. Math. Soc. (2014)

[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)

[60] T Toro, Surfaces with generalized second fundamental form in L2 are Lipschitz manifolds, J. Differential Geom. 39 (1994) 65

[61] T Toro, Geometric conditions and existence of bi-Lipschitz parameterizations, Duke Math. J. 77 (1995) 193 | DOI

[62] N T Varopoulos, L Saloff-Coste, T Coulhon, Analysis and geometry on groups, 100, Cambridge Univ. Press (1992)