On the nonexistence of bilipschitz parametrizations and geometric problems about A∞-weights.

Stephen Semmes

Revista Matemática Iberoamericana (1996)

  • Volume: 12, Issue: 2, page 337-410
  • ISSN: 0213-2230

Abstract

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How can one recognize when a metric space is bilipschitz equivalent to an Euclidean space? One should not take the abstraction of metric spaces too seriously here; subsets of Rn are already quite interesting. It is easy to generate geometric conditions which are necessary for bilipschitz equivalence, but it is not clear that such conditions should ever be sufficient. The main point of this paper is that the optimistic conjectures about the existence of bilipschitz parametrizations are wrong. In other words, there are spaces whose geometry is very similar to but still distinct from Euclidean geometry. Related questions of bilipschitz equivalence and embeddings are addressed for metric spaces obtained by deforming the Euclidean metric on Rn using an A∞ weight.

How to cite

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Semmes, Stephen. "On the nonexistence of bilipschitz parametrizations and geometric problems about A∞-weights.." Revista Matemática Iberoamericana 12.2 (1996): 337-410. <http://eudml.org/doc/39505>.

@article{Semmes1996,
abstract = {How can one recognize when a metric space is bilipschitz equivalent to an Euclidean space? One should not take the abstraction of metric spaces too seriously here; subsets of Rn are already quite interesting. It is easy to generate geometric conditions which are necessary for bilipschitz equivalence, but it is not clear that such conditions should ever be sufficient. The main point of this paper is that the optimistic conjectures about the existence of bilipschitz parametrizations are wrong. In other words, there are spaces whose geometry is very similar to but still distinct from Euclidean geometry. Related questions of bilipschitz equivalence and embeddings are addressed for metric spaces obtained by deforming the Euclidean metric on Rn using an A∞ weight.},
author = {Semmes, Stephen},
journal = {Revista Matemática Iberoamericana},
keywords = {Aplicación Lipschitziana; Espacios métricos; Variedades topológicas; Espacio euclídeo; Equivalencias geométricas; Parametrización; bi-Lipschitz equivalent to a Euclidean space; bi-Lipschitz parameterizations; deforming the Euclidean metric; weight},
language = {eng},
number = {2},
pages = {337-410},
title = {On the nonexistence of bilipschitz parametrizations and geometric problems about A∞-weights.},
url = {http://eudml.org/doc/39505},
volume = {12},
year = {1996},
}

TY - JOUR
AU - Semmes, Stephen
TI - On the nonexistence of bilipschitz parametrizations and geometric problems about A∞-weights.
JO - Revista Matemática Iberoamericana
PY - 1996
VL - 12
IS - 2
SP - 337
EP - 410
AB - How can one recognize when a metric space is bilipschitz equivalent to an Euclidean space? One should not take the abstraction of metric spaces too seriously here; subsets of Rn are already quite interesting. It is easy to generate geometric conditions which are necessary for bilipschitz equivalence, but it is not clear that such conditions should ever be sufficient. The main point of this paper is that the optimistic conjectures about the existence of bilipschitz parametrizations are wrong. In other words, there are spaces whose geometry is very similar to but still distinct from Euclidean geometry. Related questions of bilipschitz equivalence and embeddings are addressed for metric spaces obtained by deforming the Euclidean metric on Rn using an A∞ weight.
LA - eng
KW - Aplicación Lipschitziana; Espacios métricos; Variedades topológicas; Espacio euclídeo; Equivalencias geométricas; Parametrización; bi-Lipschitz equivalent to a Euclidean space; bi-Lipschitz parameterizations; deforming the Euclidean metric; weight
UR - http://eudml.org/doc/39505
ER -

Citations in EuDML Documents

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  1. Kerkko Luosto, Ultrametric spaces bi-Lipschitz embeddable in ℝ n
  2. Annalisa Baldi, Bruno Franchi, Some Remarks on Vector Potentials for Maxwell's Equations in Space-Time Carnot Groups
  3. Bruno Franchi, Marco Marchi, Raul Paolo Serapioni, Differentiability and Approximate Differentiability for Intrinsic Lipschitz Functions in Carnot Groups and a Rademacher Theorem
  4. Hervé Pajot, Plongements bilipschitziens dans les espaces euclidiens, Q -courbure et flot quasi-conforme
  5. Pierre Pansu, Plongements quasiisométriques du groupe de Heisenberg dans L p , d’après Cheeger, Kleiner, Lee, Naor
  6. Bernd Kirchheim, Francesco Serra Cassano, Rectifiability and parameterization of intrinsic regular surfaces in the Heisenberg group
  7. Franchi, Bruno, B V spaces and rectifiability for Carnot-Carathéodory metrics: an introduction

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