# zbMATH — the first resource for mathematics

Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities. (English) Zbl 0870.54031
The central objective of this paper is to give geometric criteria on a metric space $$(M,d)$$ where there exists a (rich) family of curves, starting and ending in a prescribed pair of sets. In addition, the curves of the family are controlled by a measure $$\nu$$ on the measure space $$(S,\lambda )$$ (=the indexing set of the family) in the sense that for each curve $$\gamma _p:[0,L(p)]\rightarrow M$$ of the family, ($$p\in S$$) , one has $$\mu (\gamma _n^{-1}(A))\leq a(p)$$ and $$\int_Sa(p)d\lambda (p)\leq \nu (A)$$, for each Borel set $$A\subseteq M$$, where $$a:S\rightarrow [0,\infty ]$$ is a measurable function and $$\mu$$ is the Lebesgue measure on $$\mathbb{R}$$. Generally, $$S$$ is replaced by the $$n$$-dimensional sphere $$S^n$$ and $$\nu$$ is expressed in terms of the $$n$$-dimensional Hausdorff measure on M. The conditions imposed to the space M involve asymptotic properties at infinity. The family of curves is obtained by building Lipschitz maps from M to $$S^n$$ with good topological properties. To this purpose is devoted the first of the two parts of the paper. The family of curves is also used to control the values of a function in terms of its (generalized) gradient (defined on a general metric space) and to obtain Sobolev and Poincaré inequalities (in the appendix B of the paper).

##### MSC:
 54E40 Special maps on metric spaces 57N15 Topology of the Euclidean $$n$$-space, $$n$$-manifolds ($$4 \leq n \leq \infty$$) (MSC2010) 54F45 Dimension theory in general topology 55M25 Degree, winding number 28A78 Hausdorff and packing measures
Full Text:
##### References:
 [1] P. Assouad. Espaces Métriques, Plongements, Facteurs. Thèse de Doctorat (January, 1977), Université de Paris XI, 91405 Orsay Cedex, France. [2] P. Assouad. Étude d’une dimension métrique liée à la possibilité de plongement dansR n .C. R. Acad. Sci. Paris 288 (1979), 731–734. · Zbl 0409.54020 [3] P. Assouad. Plongements Lipschitziens dansR n .Bull. Soc. Math. France 111 (1983), 429–448. · Zbl 0597.54015 [4] M. Barlow and R. Bass. Coupling and Harnack inequalities for Sierpinski carpets.Bull. Amer. Math. Soc 29 (1993), 208–212. · Zbl 0782.60017 · doi:10.1090/S0273-0979-1993-00424-5 [5] M. Bridson and G. Swarup. On Hausdorff-Gromov convergence and a theorem of Paulin.L’Enseignement Mathématique 40 (1994), 267–289. · Zbl 0846.20038 [6] R. Bott and L. Tu. Differential Forms in Algebraic Topology. Graduate Texts in Mathematics,82 Springer-Verlag, 1982. · Zbl 0496.55001 [7] J. Block and S. Weinberger. Large scale homology theories and geometry. Preprint. · Zbl 0898.55006 [8] J. Cannon. The characterization of topological manifolds of dimensionn. Proceedings I.C.M. (Helsinki, 1978), 449–454. [9] J. Cannon. Shrinking cell-like decompositions of manifolds: codimension 3.Ann. Math. (2 110 (1979), 83–112. · Zbl 0424.57007 · doi:10.2307/1971245 [10] A. Connes, M. Gromov, and H. Moscovici. Group cohomology with Lipschitz control and higher signatures.Geometric and Functional Analysis 3 (1993), 1–78. · Zbl 0789.58069 · doi:10.1007/BF01895513 [11] R. Coifman and G. Weiss. Analyse Harmonique Non-commutative sur Certains Espaces Homogènes. Lecture Notes in Math,242 Springer-Verlag, 1971. · Zbl 0224.43006 [12] R. Coifman and G. Weiss. Extensions of Hardy spaces and their use in analysis.Bull. Amer. Math. Soc 83 (1977), 569–645. · Zbl 0358.30023 · doi:10.1090/S0002-9904-1977-14325-5 [13] R. Daverman.Decompositions of Manifolds. Academic Press, 1986. · Zbl 0608.57002 [14] G. David and S. Semmes. StrongAweights, Sobolev inequalities, and quasiconformal mappings. in Analysis and Partial Differential Equations (C. Sadosky, ed.); Lecture Notes in Pure and Applied Mathematics,122 Marcel Dekker 1990 · Zbl 0752.46014 [15] G. David and S. Semmes. Quantitative rectifiability and Lipschitz mappings.Transactions Amer. Math. Soc. 337 (1993), 855–889. · Zbl 0792.49029 · doi:10.2307/2154247 [16] S. Donaldson and D. Sullivan. Quasiconformal 4-manifolds.Acta Math. 163 (1989), 181–252. · Zbl 0704.57008 · doi:10.1007/BF02392736 [17] A. Dranishnikov, S. Ferry, and S. Weinberger. Large Riemannian manifolds which are flexible. Preprint. · Zbl 1051.53035 [18] R. Edwards. The topology of manifolds and cell-like maps. Proceedings I.C.M. (Helsinki, 1978), 111–127 [19] F. T. Farrell and W. C. Hsiang. On Novikov’s conjecture for non-positively curved manifolds, I.Ann. Math. (2 113 (1981), 199–209. · Zbl 0461.57016 · doi:10.2307/1971138 [20] H. Federer.Geometric Measure Theory. Spring-Verlag, 1969. · Zbl 0176.00801 [21] S. Ferry. Counting simple-homotopy types of topological manifolds. Preprint. [22] S. Ferry. Topological finiteness theorems for manifolds in Gromov-Hausdorff space.Duke Math. J. 74 (1994), 95–106. · Zbl 0824.53040 · doi:10.1215/S0012-7094-94-07404-8 [23] S. Ferry and S. Weinberger. A coarse approach to the Novikov conjecture. Preprint. [24] S. Ferry, I. Hambleton, and E. K. Pedersen. A survey of bounded surgery theory and applications. Preprint. · Zbl 0840.57020 [25] G. Folland. Subelliptic estimates and function spaces on nilpotent Lie groups.Arkiv f. Mat. 13 (1975), 161–207. · Zbl 0312.35026 · doi:10.1007/BF02386204 [26] G. Folland and E. Stein. Estimates for the $$\bar \partial _b$$ -complex and analysis on the Heisenberg group.Comm. Pure Appl. Math. 27 (1974), 429–522. · Zbl 0293.35012 · doi:10.1002/cpa.3160270403 [27] G. Folland and E. Stein. Hardy Spaces on Homogeneous Groups.Math. Notes 28 (1982); Princeton University Press. · Zbl 0508.42025 [28] M. Freedman and F. Quinn.Topology of 4-Manifolds. Princeton University Press, 1990. · Zbl 0705.57001 [29] F. Gehring. TheL p integrability of the partial derivatives of a quasiconformal mapping.Acta Math. 130 (1973), 265–277. · Zbl 0258.30021 · doi:10.1007/BF02392268 [30] F. Gehring. The Hausdorff measure of sets which link in Euclidean space. ”Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers”, Academic Press, New York and London, 1974, p. 159–167. [31] R. Greene and P. Petersen V. Little topology, big voluem.Duke. Math. J. 67 (1992), 273–290. · Zbl 0772.53033 · doi:10.1215/S0012-7094-92-06710-X [32] K. Grove, P. Petersen V, and J.-Y. Wu. Geometric finiteness theorems via controlled topology.Invent. Math. 99 (1990), 205–213; erratum104 (1991), 221–222. · Zbl 0747.53033 · doi:10.1007/BF01234417 [33] M. Gromov.Structures Métriques pour les Variétés Riemanniennes (J. Lafontaine, P. Pansu, eds.). Cedic/Fernand Nathan, Paris, 1981. [34] M. Gromov. Groups of polynomial growth and expanding maps.Publ. Math. IHES 53 (1981), 183–215. · Zbl 0474.20018 [35] M. Gromov. Filling Riemannian manifolds.J. Differential Geometry 18 (1983), 1–147. · Zbl 0515.53037 [36] M. Gromov. Asymptotic invariants of infinite groups. Proceedings of the 1991 conference Sussex conference on Geometric Group Theory, Niblo and Koller, editors, London math. Soc. Lecture Notes182 (1993); Cambridge Univ. Press. [37] M. Gromov. Positive curvature, macroscopic dimension, spectral gaps and higher signatures. IHES preprint M/95/36, 1995. [38] P. Hajlasz and P. Koskela. Sobolev meets Poincaré.C. R. Acad. Sci. Paris 320 (1995), 1211–1215. · Zbl 0837.46024 [39] W. Hurewicz and H. Wallman.Dimension Theory. Princeton Univ. Press, 1941, 1969. · JFM 67.1092.03 [40] J. Hocking, G. Young.Topology. Addison Wesley, 1961. [41] D. Jerison. The Poincaré for vector fields satisfying Hörmander’s condition.Duke Math. J. 53 (1986), 503–523. · Zbl 0614.35066 · doi:10.1215/S0012-7094-86-05329-9 [42] R. Kirby. The Topology of 4-Manifolds. Lecture Notes in Math, vol. 1374. Springer-Verlag, 1989. · Zbl 0668.57001 [43] W. Massey.A Basic Course in Algebraic Topology. Springer-Verlag, 1991. · Zbl 0725.55001 [44] J. Milnor.Topology from the Differentiable Viewpoint. University of Virginia Press, Charlottesville, 1965. · Zbl 0136.20402 [45] E. E. Moise.Geometric Topology in Dimensions 2and 3. Springer-Verlag, 1977. · Zbl 0349.57001 [46] L. Nirenberg.Topics in Nonlinear Functional Analysis. Courant Institute of Mathematical Sciences, 1974. · Zbl 0286.47037 [47] P. Petersen V. A finiteness theorem for metric spaces.J. Diff. Geom. 31 (1990). 387–395. · Zbl 0696.55005 [48] P. Petersen V. Gromov-Hausdorff convergence of metric spaces.Proc. Sym. Pure Math. 54 Part 3 (1993), 489–504; Amer. Math. Soc. · Zbl 0788.53037 [49] W. Rudin.Real and Complex Analysis. McGraw-Hill, 1987. · Zbl 0925.00005 [50] S. Semmes. Differentiable function theory on hypersurfaces inR n (without bounds on their smoothness).Indiana Math. J. 39 (1990), 985–1004. · Zbl 0796.42013 · doi:10.1512/iumj.1990.39.39047 [51] S. Semmes. Bilipschitz mappings and strongAweights.Ann. Acad. Sci. Fenn. A I 18 (1993), 211–248. · Zbl 0742.46010 [52] S. Semmes. Finding structure in sets with little smoothness. Proc. I.C.M. (Zürich, 1994), Birkhäuser, 1995, p875–885. · Zbl 0897.28003 [53] S. Semmes.On the nonexistence of bilipschitz parameterizations and geometric problems about Aweights. Revista Matemática Iberoamericana, 1996. · Zbl 0858.46017 [54] S. Semmes.Good metric spaces without good parameterizations. Revista Matemática Iberoamericana, vol 12, p. 187–275, 1996. · Zbl 0854.57018 [55] S. Semmes. Some remarks about metric spaces, spherical mappings, functions and their derivatives (to appear); Publications Mathemàtiques. · Zbl 0927.46021 [56] L. Siebenmann and D. Sullivan. On complexes that are Lipschitz manifolds. ”Geometric Topology” (J. Cantrell, ed.). Academic Press, 1979, 503–525. · Zbl 0478.57008 [57] E. M. Stein.Singular Integrals and Differentiability Properties of Functions. Princeton University Press, 1970. · Zbl 0207.13501 [58] E. M. Stein.Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, 1993. · Zbl 0821.42001 [59] D. Sullivan. Hyperbolic geometry and homeomorphisms. ”Geometric Topology” (J. Cantrell, ed.). Academic Press, 1979, 543–555 [60] J. Väisälä. Quasisymmetric embeddings in Euclidean spaces.Trans. Amer. Math. Soc. 264 (1981), 191–204. · Zbl 0456.30018 · doi:10.1090/S0002-9947-1981-0597876-7 [61] J. Väisälä. Metric duality in Euclidean spaces. Preprint. [62] F. Warner.Foundations of Differentiable Manifolds and Lie Groups. Scott Foresman and Co, 1971. · Zbl 0241.58001 [63] H. Whitney. On maps of ann-sphere into anothern-sphere.Duke Math. J. 3 (1937), 46–50. · JFM 63.1162.01 · doi:10.1215/S0012-7094-37-00305-3 [64] G. Yu. Zero-in-the-spectrum conjecture, positive scalar curvature and asymptotic dimension. Preprint. · Zbl 0889.58082
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.