On the locally branched Euclidean metric gauge
type: Item
instance of: scholarly article
title: On the locally branched Euclidean metric gauge
zbMATH Open document ID: 1019.58002
DOI: 10.1215/S0012-7094-02-11412-4
author: Juha Heinonen
author: Dennis Sullivan
published in: Duke Mathematical Journal
publication date: 2002-10-27T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.dmj/1087575355
review text: A metric gauge on a set is a maximal collection of metrics on the set such that the identity map between any two metrics from the collection is locally bi-Lipschitz. The authors present a characterization for metric gauges that are locally ``branched Euclidean'' and discuss an obstruction to removing the branching. The \(n\)-dimensional gauges that are embeddable in a finite-dimensional Euclidean space and whose local cohomology groups in dimensions \((n-1)\) and higher are similar to those of an \(n\)-manifold are considered. The approach is to stipulate enough structure so that one can consider differential Whitney 1-forms on the gauge together with an orientation on the measurable cotangent bundle that is compatible with a chosen local topological orientation. An \(n\)-tuple \(\rho = (\rho_1,\dots, \rho_n)\) of locally defined 1-forms on an \(n\)-dimensional gauge is called a (local) Cartan-Whitney presentation of the gauge if \[ \text{ess inf} \ast (\rho_1\wedge\dots\wedge \rho_n) > 0. \tag{1} \] It is proved that if the gauge supports, in addition, a Poincaré inequality, then each (local) Cartan-Whitney presentation \(\rho\) determines a positive integer-valued function Res\((\rho, \cdot)\), the residue of the presentation, such that the metric gauge is locally Euclidean at a point \(\rho\) iff the residue (of some presentation) satisfies Res\((\rho, p) =1\). Moreover, for each presentation \(\rho\), the residue function Res\((\rho,\cdot)\) assumes the value 1 on a dense open set of full measure with complement at most \((n-2)\)-dimensional. In particular, the existence of local Cartan-Whitney presentations implies that the gauge is locally Euclidean almost everywhere. The main ingredient of the proof is a general form of a theorem of Reshetnyak. It is shown that the map \[ x\mapsto f(x)=\int_{[p,x]}(\rho_1,\dots,\rho_n),\tag{2} \] defined through integration of the 1-forms \(\rho_1,\dots,\rho_n\) as in (1), defines a Lipschitz branched cover into \(\mathbb R^n…
Mathematics Subject Classification ID: 58A99
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 30C65
zbMATH DE Number: 1820917
zbMATH Keywords: metric gauge
zbMATH Keywords: locally bi-Lipschitz metrics
zbMATH Keywords: locally branched Euclidean metric gauges
zbMATH Keywords: obstruction to removing the branching
zbMATH Keywords: \(n\)-dimensional gauges
zbMATH Keywords: cohomology manifolds
zbMATH Keywords: metric orientation
zbMATH Keywords: Sobolev classes
zbMATH Keywords: gauge Whitney 1-forms
zbMATH Keywords: differential Whitney 1-forms
zbMATH Keywords: Stokes cycles
zbMATH Keywords: local cohomology groups
zbMATH Keywords: measurable cotangent bundle
zbMATH Keywords: Cartan-Whitney presentation of the gauge
zbMATH Keywords: residue of presentation
zbMATH Keywords: Lipschitz branched cover
zbMATH Keywords: differential analysis and nonlinear potential theory on metric measure spaces with Poincaré inequality
MaRDI profile type: Publication
cites work: Plongements lipschitziens dans ${\bbfR}\sp n$
cites work: On the Construction of Branched Coverings of Low-Dimensional Manifolds
cites work: The recognition problem: What is a topological manifold?
cites work: Q3860516
cites work: Differentiability of Lipschitz functions on metric measure spaces
cites work: Q5337813
cites work: Q5337814
cites work: Q4368752
cites work: Q3865166
cites work: Q5565773
cites work: Definitions of Sobolev classes on metric spaces
cites work: Sobolev met Poincaré
cites work: BLD-mappings in \(W^{2,2}\) are locally invertible
cites work: Q4693180
cites work: Sobolev mappings with integrable dilatations
cites work: Quasiconformal maps in metric spaces with controlled geometry
cites work: Quasiregular maps \(\mathbb{S}^3\to\mathbb{S}^3\) with wild branch sets
cites work: Geometric branched covers between generalized manifolds
cites work: Q5837247
cites work: Q4400436
cites work: PLANE WITH $A_{\infty}$ -WEIGHTED METRIC NOT BILIPSCHITZ EMBEDDABLE TO ${\bb R}^n$
cites work: Elliptic equations and maps of bounded length distortion
cites work: Space mappings with bounded distortion
cites work: Q3992482
cites work: Q3141918
cites work: Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities
cites work: Good metric spaces without good parameterizations
cites work: On the nonexistence of bilipschitz parameterizations and geometric problems about \(A_ \infty\)-weights
cites work: Bilipschitz embeddings of metric spaces into Euclidean spaces
cites work: Q4528988
cites work: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces
cites work: Q3935870
cites work: Q4266113
cites work: The Extension of Interiority, With Some Applications
cites work: Surfaces with generalized second fundamental form in \(L^ 2\) are Lipschitz manifolds
cites work: Geometric conditions and existence of bi-Lipschitz parameterizations
cites work: Q5520541
cites work: Q5520542
cites work: Algebraic Topology and Integration Theory
cites work: Q3253193
cites work: Q5799228
description: scientific article; zbMATH DE number 1820917