On the locally branched Euclidean metric gauge
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 \(\mathbb R^n\), with the property that \[ \lim_{y\to x}\inf_{y\neq x}\frac{|f(x)-f(y)|}{d(x, y)}\geq c > 0 \] for all \(x\) and for some \(c > 0\) independent of \(x\). The residue Res\((\rho,p)\) is the local index of the map (2) at \(p\). All this is made more precise in the main theorem of the paper. To prove the theorem, the authors make use of the recent advances in differential analysis and nonlinear potential theory on metric measure spaces with Poincaré inequality. The metric gauges that admit local Cartan-Whitney presentations need not be manifolds in general, and even if they are manifolds they need not be locally Euclidean. But they are always branched Euclidean.
locally branched Euclidean metric gauges
Cartan-Whitney presentation of the gauge
differential analysis and nonlinear potential theory on metric measure spaces with Poincaré inequality
scientific article; zbMATH DE number 1820917
scientific article