Infinitesimal splitting for spaces with thick curve families and Euclidean embeddings
Annales de l'Institut Fourier, Online first, 44 p.

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× k for some k1. We also provide applications to conformal dimension and give new proofs of some previously known non-embedding results.

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× k pour un certain k1. On donne aussi des applications à la dimension conforme et de nouvelles preuves de certains résultats de non plongement déjà connus.

Received:
Revised:
Accepted:
Online First:
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
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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.

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