Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings
[Décomposition infinitésimale pour des espaces munis de familles de courbes épaisses et plongements euclidiens]
David, Guy C.1 ; Eriksson-Bique, Sylvester2
1 Department of Mathematical Sciences, Ball State University, Muncie, IN 47306
2 Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, Jyväskylä, Finland
Annales de l'Institut Fourier, Online first, 44 p.

On étudie des espaces métriques mesurés qui possèdent des familles “épaisses” de courbes rectifiables ou de fragments de courbes, sous la forme de représentations d’Alberti ou de familles de courbes de module strictement positif. On montre que de tels espaces ne possèdent pas de plongement bi-lipschitzien dans un espace euclidien, sauf s’ils admettent une “décomposition infinitésimale” : leurs espaces tangents sont bi-lipschitz équivalents à des produits d’espaces de la forme Z×ℝ k pour un certain k≥1. On donne aussi des applications à la dimension conforme et de nouvelles preuves de certains résultats de non plongement déjà connus.

We study metric measure spaces that admit “thick” families of rectifiable curves or curve fragments, in the form of Alberti representations or curve families of positive modulus. We show that such spaces cannot be bi-Lipschitz embedded into any Euclidean space unless they admit some “infinitesimal splitting”: their tangent spaces are bi-Lipschitz equivalent to product spaces of the form Z×ℝ k for some k≥1. We also provide applications to conformal dimension and give new proofs of some previously known non-embedding results.

Reçu le :
Révisé le :
Accepté le :
Première publication :
DOI : 10.5802/aif.3606
Classification : 30L05, 53C23, 49J52
Keywords: bi-Lipschitz embedding, modulus, conformal dimension, Alberti representation
Mot clés : plongement bi-lipschitz, module, dimension conforme, représentation d’Alberti
David, Guy C. 1 ; Eriksson-Bique, Sylvester 2

1 Department of Mathematical Sciences, Ball State University, Muncie, IN 47306
2 Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, Jyväskylä, Finland 
@unpublished{AIF_0__0_0_A57_0,
     author = {David, Guy C. and Eriksson-Bique, Sylvester},
     title = {Infinitesimal splitting for spaces with thick curve families and {Euclidean} embeddings},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     year = {2024},
     doi = {10.5802/aif.3606},
     language = {en},
     note = {Online first},
}
TY  - UNPB
AU  - David, Guy C.
AU  - Eriksson-Bique, Sylvester
TI  - Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings
JO  - Annales de l'Institut Fourier
PY  - 2024
PB  - Association des Annales de l’institut Fourier
N1  - Online first
DO  - 10.5802/aif.3606
LA  - en
ID  - AIF_0__0_0_A57_0
ER  - 
%0 Unpublished Work
%A David, Guy C.
%A Eriksson-Bique, Sylvester
%T Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings
%J Annales de l'Institut Fourier
%D 2024
%I Association des Annales de l’institut Fourier
%Z Online first
%R 10.5802/aif.3606
%G en
%F AIF_0__0_0_A57_0
David, Guy C.; Eriksson-Bique, Sylvester. Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings. Annales de l'Institut Fourier, Online first, 44 p.

[1] Alberti, Giovanni Rank one property for derivatives of functions with bounded variation, Proc. R. Soc. Edinb., Sect. A, Math., Volume 123 (1993) no. 2, pp. 239-274 | DOI | MR | Zbl

[2] Alberti, Giovanni; Csörnyei, Marianna; Preiss, David Structure of null sets in the plane and applications, European Congress of Mathematics, European Mathematical Society, 2005, pp. 3-22 | MR | Zbl

[3] Alberti, Giovanni; Marchese, Andrea On the differentiability of Lipschitz functions with respect to measures in the Euclidean space, Geom. Funct. Anal., Volume 26 (2016) no. 1, pp. 1-66 | DOI | MR | Zbl

[4] Ambrosio, Luigi; Di Marino, Simone; Savaré, Giuseppe On the duality between p-modulus and probability measures, J. Eur. Math. Soc., Volume 17 (2015) no. 8, pp. 1817-1853 | DOI | MR | Zbl

[5] Ambrosio, Luigi; Kirchheim, Bernd Rectifiable sets in metric and Banach spaces, Math. Ann., Volume 318 (2000) no. 3, pp. 527-555 | DOI | MR | Zbl

[6] Ambrosio, Luigi; Kleiner, Bruce; Le Donne, Enrico Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane, J. Geom. Anal., Volume 19 (2009) no. 3, pp. 509-540 | DOI | MR | Zbl

[7] Ambrosio, Luigi; Tilli, Paolo Topics on analysis in metric spaces, Oxford Lecture Series in Mathematics and its Applications, 25, Oxford University Press, 2004, viii+133 pages | MR

[8] Bate, David Structure of measures in Lipschitz differentiability spaces, J. Am. Math. Soc., Volume 28 (2015) no. 2, pp. 421-482 | DOI | MR | Zbl

[9] Bishop, Christopher J.; Peres, Yuval Fractals in probability and analysis, Cambridge Studies in Advanced Mathematics, 162, Cambridge University Press, 2017, ix+402 pages | DOI | MR

[10] Bonk, Mario Quasiconformal geometry of fractals, International Congress of Mathematicians. Vol. II, European Mathematical Society, 2006, pp. 1349-1373 | MR | Zbl

[11] Bonk, Mario; Kleiner, Bruce Conformal dimension and Gromov hyperbolic groups with 2-sphere boundary, Geom. Topol., Volume 9 (2005), pp. 219-246 | DOI | MR | Zbl

[12] Bonk, Mario; Merenkov, Sergei Quasisymmetric rigidity of square Sierpiński carpets, Ann. Math., Volume 177 (2013) no. 2, pp. 591-643 | DOI | MR | Zbl

[13] Bourdon, Marc; Pajot, Hervé Cohomologie l p et espaces de Besov, J. Reine Angew. Math., Volume 558 (2003), pp. 85-108 | DOI | MR | Zbl

[14] Burago, Dmitri; Burago, Yuriĭ; Ivanov, Sergei A course in metric geometry, Graduate Studies in Mathematics, 33, American Mathematical Society, 2001, xiv+415 pages | DOI | MR

[15] Capogna, Luca; Danielli, Donatella; Pauls, Scott D.; Tyson, Jeremy T. An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, Progress in Mathematics, 259, Birkhäuser, 2007, xvi+223 pages | MR

[16] Cheeger, Jeff Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal., Volume 9 (1999) no. 3, pp. 428-517 | DOI | MR | Zbl

[17] Cheeger, Jeff; Kleiner, Bruce On the differentiability of Lipschitz maps from metric measure spaces to Banach spaces, Inspired by S. S. Chern (Nankai Tracts in Mathematics), Volume 11, World Scientific, 2006, pp. 129-152 | DOI | MR | Zbl

[18] Cheeger, Jeff; Kleiner, Bruce Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodým property, Geom. Funct. Anal., Volume 19 (2009) no. 4, pp. 1017-1028 | DOI | MR | Zbl

[19] Cheeger, Jeff; Kleiner, Bruce; Schioppa, Andrea Infinitesimal structure of differentiability spaces, and metric differentiation, Anal. Geom. Metr. Spaces, Volume 4 (2016) no. 1, pp. 104-159 | DOI | MR | Zbl

[20] David, Guy; Semmes, Stephen Fractured fractals and broken dreams, Oxford Lecture Series in Mathematics and its Applications, 7, Clarendon Press, 1997, x+212 pages | DOI

[21] David, Guy C. Tangents and rectifiability of Ahlfors regular Lipschitz differentiability spaces, Geom. Funct. Anal., Volume 25 (2015) no. 2, pp. 553-579 | DOI | MR | Zbl

[22] David, Guy C.; Eriksson-Bique, Sylvester Regular mappings and non-existence of bi-Lipschitz embeddings for slit carpets, Adv. Math., Volume 364 (2020), 107047, 14 pages | DOI | MR | Zbl

[23] David, Guy C.; Kinneberg, Kyle Rigidity for convex-cocompact actions on rank-one symmetric spaces, Geom. Topol., Volume 22 (2018) no. 5, pp. 2757-2790 | DOI | MR | Zbl

[24] Durand-Cartagena, Estibalitz; Eriksson-Bique, Sylvester; Korte, Riikka; Shanmugalingam, Nageswari Equivalence of two BV classes of functions in metric spaces, and existence of a Semmes family of curves under a 1-Poincaré inequality, Adv. Calc. Var., Volume 14 (2021) no. 2, pp. 231-245 | DOI | MR | Zbl

[25] Hajłasz, Piotr Non embedding of the Heisenberg group MathOverflow, https://mathoverflow.net/q/297806 (version: 2018-04-16)

[26] Heinonen, Juha Lectures on analysis on metric spaces, Universitext, Springer, 2001, x+140 pages | DOI

[27] Heinonen, Juha Nonsmooth calculus, Bull. Am. Math. Soc., Volume 44 (2007) no. 2, pp. 163-232 | DOI | MR | Zbl

[28] Heinonen, Juha; Koskela, Pekka Quasiconformal maps in metric spaces with controlled geometry, Acta Math., Volume 181 (1998) no. 1, pp. 1-61 | DOI | MR | Zbl

[29] Heinonen, Juha; Koskela, Pekka; Shanmugalingam, Nageswari; Tyson, Jeremy T. Sobolev spaces on metric measure spaces. An approach based on upper gradients, New Mathematical Monographs, 27, Cambridge University Press, 2015, xii+434 pages | DOI | MR

[30] Heinonen, Juha; Semmes, Stephen Thirty-three yes or no questions about mappings, measures, and metrics, Conform. Geom. Dyn., Volume 1 (1997), pp. 1-12 | DOI | MR | Zbl

[31] Honzlová Exnerová, Vendula; Kalenda, Ondřej F. K.; Malý, Jan; Martio, Olli Plans on measures and AM-modulus, J. Funct. Anal., Volume 281 (2021) no. 10, 109205, 35 pages | MR | Zbl

[32] Järvenpää, Esa; Järvenpää, Maarit; Rogovin, Kevin; Rogovin, Sari; Shanmugalingam, Nageswari Measurability of equivalence classes and MEC p -property in metric spaces, Rev. Mat. Iberoam., Volume 23 (2007) no. 3, pp. 811-830 | DOI | MR | Zbl

[33] Keith, Stephen Modulus and the Poincaré inequality on metric measure spaces, Math. Z., Volume 245 (2003) no. 2, pp. 255-292 | DOI | MR | Zbl

[34] Keith, Stephen A differentiable structure for metric measure spaces, Adv. Math., Volume 183 (2004) no. 2, pp. 271-315 | DOI | MR | Zbl

[35] Keith, Stephen; Laakso, Tomi Conformal Assouad dimension and modulus, Geom. Funct. Anal., Volume 14 (2004) no. 6, pp. 1278-1321 | DOI | MR | Zbl

[36] Kinneberg, Kyle Conformal dimension and boundaries of planar domains, Trans. Am. Math. Soc., Volume 369 (2017) no. 9, pp. 6511-6536 | DOI | MR | Zbl

[37] Kirchheim, Bernd Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Am. Math. Soc., Volume 121 (1994) no. 1, pp. 113-123 | DOI | MR | Zbl

[38] Kwapisz, Jaroslaw Conformal dimension via p-resistance: Sierpiński carpet, Ann. Acad. Sci. Fenn., Math., Volume 45 (2020) no. 1, pp. 3-51 | DOI | MR | Zbl

[39] Laakso, Tomi Ahlfors Q-regular spaces with arbitrary Q>1 admitting weak Poincaré inequality, Geom. Funct. Anal., Volume 10 (2000) no. 1, pp. 111-123 | DOI | MR | Zbl

[40] Le Donne, Enrico Metric spaces with unique tangents, Ann. Acad. Sci. Fenn., Math., Volume 36 (2011) no. 2, pp. 683-694 | DOI | MR | Zbl

[41] Le Donne, Enrico A primer on Carnot groups: homogenous groups, Carnot–Carathéodory spaces, and regularity of their isometries, Anal. Geom. Metr. Spaces, Volume 5 (2017) no. 1, pp. 116-137 | DOI | MR | Zbl

[42] Mackay, John M.; Tyson, Jeremy T. Conformal dimension. Theory and application, University Lecture Series, 54, American Mathematical Society, 2010, xiv+143 pages | DOI | MR

[43] Mattila, Pertti Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability, Cambridge Studies in Advanced Mathematics, 44, Cambridge University Press, 1995, xii+343 pages | DOI | MR

[44] Mattila, Pertti Measures with unique tangent measures in metric groups, Math. Scand., Volume 97 (2005) no. 2, pp. 298-308 | DOI | MR | Zbl

[45] Merenkov, Sergei A Sierpiński carpet with the co-Hopfian property, Invent. Math., Volume 180 (2010) no. 2, pp. 361-388 | DOI | MR | Zbl

[46] Nagata, Jun-iti Modern dimension theory, Sigma Series in Pure Mathematics, 2, Heldermann Verlag, 1983, ix+284 pages | MR | Zbl

[47] Ostrovskii, Mikhail I. On metric characterizations of the Radon–Nikodým and related properties of Banach spaces, J. Topol. Anal., Volume 6 (2014) no. 3, pp. 441-464 | DOI | MR | Zbl

[48] Ostrovskii, Mikhail I. Radon–Nikodým property and thick families of geodesics, J. Math. Anal. Appl., Volume 409 (2014) no. 2, pp. 906-910 | DOI | MR | Zbl

[49] Pansu, Pierre Dimension conforme et sphère à l’infini des variétés à courbure négative, Ann. Acad. Sci. Fenn., Math., Volume 14 (1989) no. 2, pp. 177-212 | DOI | MR | Zbl

[50] Pansu, Pierre Métriques de Carnot–Carathéodory et quasiisométries des espaces symétriques de rang un, Ann. Math., Volume 129 (1989) no. 1, pp. 1-60 | DOI | MR | Zbl

[51] Preiss, David Geometry of measures in R n : distribution, rectifiability, and densities, Ann. Math., Volume 125 (1987) no. 3, pp. 537-643 | DOI | MR | Zbl

[52] Rudin, Walter Function theory in the unit ball of ℂ n , Classics in Mathematics, Springer, 2008, xiv+436 pages (reprint of the 1980 edition) | MR

[53] Schioppa, Andrea Derivations and Alberti representations, Adv. Math., Volume 293 (2016), pp. 436-528 | DOI | MR | Zbl

[54] Semmes, Stephen On the nonexistence of bi-Lipschitz parameterizations and geometric problems about A ∞ -weights, Rev. Mat. Iberoam., Volume 12 (1996) no. 2, pp. 337-410 | DOI | MR | Zbl

[55] Sion, Maurice On general minimax theorems, Pac. J. Math., Volume 8 (1958) no. 1, pp. 171-176 | DOI | MR | Zbl

[56] Tyson, Jeremy T. Sets of minimal Hausdorff dimension for quasiconformal maps, Proc. Am. Math. Soc., Volume 128 (2000) no. 11, pp. 3361-3367 | DOI | MR | Zbl

[57] Waterhouse, William C. Uniform Convergence and Graph Convergence, Am. Math. Mon., Volume 83 (1976) no. 8, pp. 641-643 | DOI | MR | Zbl

[58] Weaver, Nik Lipschitz algebras and derivations. II. Exterior differentiation, J. Funct. Anal., Volume 178 (2000) no. 1, pp. 64-112 | DOI | MR | Zbl

[59] Ziemer, William P. Extremal length and p-capacity, Mich. Math. J., Volume 16 (1969), pp. 43-51 | MR | Zbl

Cité par Sources :