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On the locally branched Euclidean metric gauge. (English) Zbl 1019.58002

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.

MSC:

58A99 General theory of differentiable manifolds
57M12 Low-dimensional topology of special (e.g., branched) coverings
30C65 Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations
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[1] J. W. Alexander, Note on Riemannian spaces , Bull. Amer. Math. Soc. 26 (1920), 370–372. · JFM 47.0529.02 · doi:10.1090/S0002-9904-1920-03319-7
[2] P. Assouad, Plongements lipschitziens dans \(\mathbf{R}^n\) , Bull. Soc. Math. France 111 (1983), 429–448. · Zbl 0597.54015
[3] I. Berstein and A. L. Edmonds, On the construction of branched coverings of low-dimensional manifolds , Trans. Amer. Math. Soc. 247 (1979), 87–124. JSTOR: · Zbl 0359.55001 · doi:10.2307/1998776
[4] J. Cannon, The recognition problem: What is a topological manifold? Bull. Amer. Math. Soc. 84 (1978), 832–866. · Zbl 0418.57005 · doi:10.1090/S0002-9904-1978-14527-3
[5] –. –. –. –., “The characterization of topological manifolds of dimension \(n\geq5\)” in Proceedings of the International Congress of Mathematicians (Helsinki, 1978) , Acad. Sci. Fennica, Helsinki, 1980, 449–454. · Zbl 0425.57002
[6] J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces , Geom. Funct. Anal. 9 (1999), 428–517. · Zbl 0942.58018 · doi:10.1007/s000390050094
[7] A. V. Černavskiĭ, Finite-to-one open mappings of manifolds (in Russian), Mat. Sb. (N.S.) 65 (107) (1964), 357–369. · Zbl 0129.15003
[8] –. –. –. –., Addendum to the paper “Finite-to-one open mappings of manifolds (in Russian), Mat. Sb. (N.S.) 66 (108) (1965), 471–472. · Zbl 0129.15101
[9] G. David and S. Semmes, Fractured Fractals and Broken Dreams: Self-similar Geometry through Metric and Measure , Oxford Lecture Ser. Math. Appl. 7 , Oxford Univ. Press, New York, 1997. · Zbl 0887.54001
[10] R. D. Edwards, “The topology of manifolds and cell-like maps” in Proceedings of the International Congress of Mathematicians (Helsinki, 1978) , Acad. Sci. Fennica, Helsinki, 1980, 111–127. · Zbl 0428.57004
[11] H. Federer, Geometric Measure Theory , Grundlehren Math. Wiss. 153 , Springer, New York, 1969. · Zbl 0176.00801
[12] B. Franchi, P. Hajłasz, and P. Koskela, Definitions of Sobolev classes on metric spaces , Ann. Inst. Fourier (Grenoble) 49 (1999), 1903–1924. · Zbl 0938.46037 · doi:10.5802/aif.1742
[13] P. Hajłasz and P. Koskela, Sobolev met Poincaré , Mem. Amer. Math. Soc. 145 (2000), no. 688. · Zbl 0954.46022
[14] J. Heinonen and T. Kilpeläinen, \(BLD\)-mappings in \(W^{2,2}\) are locally invertible , Math. Ann. 318 (2000), 391–396. · Zbl 0967.30015 · doi:10.1007/s002080000129
[15] J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations , Oxford Math. Monogr., Oxford Univ. Press, New York, 1993. · Zbl 0780.31001
[16] J. Heinonen and P. Koskela, Sobolev mappings with integrable dilatations , Arch. Rational Mech. Anal. 125 (1993), 81–97. · Zbl 0792.30016 · doi:10.1007/BF00411478
[17] –. –. –. –., Quasiconformal maps in metric spaces with controlled geometry , Acta Math. 181 (1998), 1–61. · Zbl 0915.30018 · doi:10.1007/BF02392747
[18] J. Heinonen and S. Rickman, Quasiregular maps \(\mathbf{S}^3\to \mathbf{S}^3\) with wild branch sets , Topology 37 (1998), 1–24. · Zbl 0895.30016 · doi:10.1016/S0040-9383(97)00015-3
[19] –. –. –. –., Geometric branched covers between generalized manifolds , Duke Math. J 113 (2002), 465–529. · Zbl 1017.30023 · doi:10.1215/S0012-7094-02-11333-7
[20] W. Hurewicz and H. Wallman, Dimension Theory , Princeton Math. Ser. 4 , Princeton Univ. Press, Princeton, 1941. · Zbl 0060.39808
[21] T. Kilpeläinen, A remark on the uniqueness of quasi continuous functions , Ann. Acad. Sci. Fenn. Math. 23 (1998), 261–262. · Zbl 0919.31006
[22] T. J. Laakso, Plane with \(A_{\infty}\)-weighted metric not bilipschitz embeddable to \(\mathbf{R}^{n}\) , to appear in Bull. London Math. Soc. · Zbl 1029.30014 · doi:10.1112/S0024609302001200
[23] O. Martio and J. Väisälä, Elliptic equations and maps of bounded length distortion , Math. Ann. 282 (1988), 423–443. · Zbl 0632.35021 · doi:10.1007/BF01460043
[24] Yu. G. Reshetnyak, Spatial mappings with bounded distortion (in Russian), Sibirsk. Mat. Ž. 8 (1967), 629–658.; English translation in Sib. Math. J. 8 (1967), 466–487. · Zbl 0167.06601 · doi:10.1007/BF02196429
[25] ——–, Space Mappings with Bounded Distortion , trans. H. H. McFaden, Trans. Math. Monogr. 73 , Amer. Math. Soc., Providence, 1989. · Zbl 0667.30018
[26] S. Rickman, Quasiregular Mappings , Ergeb. Math. Grenzgeb. (3) 26 , Springer, Berlin, 1993. · Zbl 0816.30017
[27] S. Semmes, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities , Selecta Math. (N.S.) 2 (1996), 155–295. · Zbl 0870.54031 · doi:10.1007/BF01587936
[28] –. –. –. –., Good metric spaces without good parameterizations , Rev. Mat. Iberoamericana 12 (1996), 187–275. · Zbl 0854.57018 · doi:10.4171/RMI/198
[29] –. –. –. –., On the nonexistence of bi-Lipschitz parameterizations and geometric problems about \(A_{\infty}\)-weights , Rev. Mat. Iberoamericana 12 (1996), 337–410. · Zbl 0858.46017 · doi:10.4171/RMI/201
[30] –. –. –. –., Bilipschitz embeddings of metric spaces into Euclidean spaces , Publ. Mat. 43 (1999), 571–653. · Zbl 1131.30337 · doi:10.5565/PUBLMAT_43299_06
[31] ——–, Some Novel Types of Fractal Geometry , Oxford Math. Monogr., Oxford Univ. Press, New York, 2001. \CMP1 815 356 · Zbl 0970.28001
[32] N. Shanmugalingam, Newtonian spaces: An extension of Sobolev spaces to metric measure spaces , Rev. Mat. Iberoamericana 16 (2000), 243–279. · Zbl 0974.46038 · doi:10.4171/RMI/275
[33] L. Siebenmann and D. Sullivan, “On complexes that are Lipschitz manifolds” in Geometric Topology (Athens, Ga., 1977) , ed. J. C. Cantrell, Academic Press, New York, 1979, 503–525. · Zbl 0478.57008
[34] D. Sullivan,“The exterior \(d\), the local degree, and smoothability” in Prospects of Topology , ed. F. Quinn, Ann. of Math. Stud. 138 , Princeton Univ. Press, Princeton, 1995. · Zbl 0926.57027
[35] C. J. Titus and G. S. Young, The extension of interiority, with some applications , Trans. Amer. Math. Soc. 103 (1962), 329–340. JSTOR: · Zbl 0113.38001 · doi:10.2307/1993663
[36] T. Toro, Surfaces with generalized second fundamental form in \(L^2\) are Lipschitz manifolds , J. Differential Geom. 39 (1994), 65–101. · Zbl 0806.53020
[37] –. –. –. –., Geometric conditions and existence of bi-Lipschitz parametrizations , Duke Math. J. 77 (1995), 193–227. · Zbl 0847.42011 · doi:10.1215/S0012-7094-95-07708-4
[38] J. Väisälä, Minimal mappings in euclidean spaces , Ann. Acad. Sci. Fenn. Ser. A I (1965), no. 366. · Zbl 0144.22103
[39] ——–, Discrete open mappings on manifolds , Ann. Acad. Sci. Fenn. Ser. A I (1966), no. 392. · Zbl 0144.22202
[40] H. Whitney, Algebraic topology and integration theory , Proc. Natl. Acad. Sci. U.S.A. 33 (1947), 1–6. JSTOR: · Zbl 0029.42002 · doi:10.1073/pnas.33.1.1
[41] ——–, Geometric Integration Theory , Princeton Univ. Press, Princeton, 1957. · Zbl 0083.28204
[42] R. Wilder, Topology of Manifolds , Amer. Math. Soc. Colloq. Publ. 32 , Amer. Math. Soc., New York, 1949. · Zbl 0039.39602
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