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Metric differentiation, monotonicity and maps to L 1

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Abstract

This is one of a series of papers on Lipschitz maps from metric spaces to L 1. Here we present the details of results which were announced in Cheeger and Kleiner (Ann. Math., 2006, to appear, arXiv:math.MG/0611954, Sect. 1.8): a new approach to the infinitesimal structure of Lipschitz maps into L 1, and, as a first application, an alternative proof of the main theorem of Cheeger and Kleiner (Ann. Math., 2006, to appear, arXiv:math.MG/0611954), that the Heisenberg group does not admit a bi-Lipschitz embedding in L 1. The proof uses the metric differentiation theorem of Pauls (Commun. Anal. Geom. 9(5):951–982, 2001) and the cut metric description in Cheeger and Kleiner (Ann. Math., 2006, to appear, arXiv:math.MG/0611954) to reduce the nonembedding argument to a classification of monotone subsets of the Heisenberg group. A quantitative version of this classification argument is used in our forthcoming joint paper with Assaf Naor (Cheeger et al. in arXiv:0910.2026, 2009).

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References

  1. Ambrosio, L.: Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces. Adv. Math. 159(1), 51–67 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ambrosio, L.: Fine properties of sets of finite perimeter in doubling metric measure spaces. Set-Valued Anal. 10(2–3), 111–128 (2002). Calculus of variations, nonsmooth analysis and related topics

    Article  MATH  MathSciNet  Google Scholar 

  3. Assouad, P.: Plongements isométriques dans L 1: aspect analytique. In: Initiation Seminar on Analysis: G. Choquet, M. Rogalski, J. Saint-Raymond, 19th Year: 1979/1980. Publ. Math. Univ. Pierre et Marie Curie, vol. 41, p. Exp. No. 14, 23. Univ. Paris VI, Paris (1980)

    Google Scholar 

  4. Aumann, Y., Rabani, Y.: An O(log k) approximate min-cut max-flow theorem and approximation algorithm. SIAM J. Comput. 27(1), 291–301 (1998) (electronic)

    Article  MATH  MathSciNet  Google Scholar 

  5. Benyamini, Y., Lindenstrauss, J.: Geometric Nonlinear Functional Analysis. Vol. 1. American Mathematical Society Colloquium Publications, vol. 48. American Mathematical Society, Providence (2000)

    Google Scholar 

  6. Bourgain, J.: On Lipschitz embedding of finite metric spaces in Hilbert space. Isr. J. Math. 52(1–2), 46–52 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chakrabarti, A., Jaffe, A., Lee, J.R., Vincent, J.: Embeddings of topological graphs: Lossy invariants, linearization, and 2-sums. In: 49th Annual Symposium on Foundations of Computer Science, pp. 761–770 (2008)

  8. Cheeger, J.: Differentiability of Lipschitz functions on metric measure spaces. Geom. Funct. Anal. 9(3), 428–517 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  9. Cheeger, J., Kleiner, B.: Inverse limits of metric graphs, bi-Lipschitz embeddings in L 1 and PI spaces (in preparation)

  10. Cheeger, J., Kleiner, B.: Metric differentiation for PI spaces (in preparation)

  11. Cheeger, J., Kleiner, B.: Differentiating maps to L 1 and the geometry of BV functions. Ann. Math. (2006, to appear). arXiv:math.MG/0611954

  12. Cheeger, J., Kleiner, B.: Differentiability of Lipschitz maps from metric measure spaces to Banach spaces with the Radon-Nikodým property. Geom. Funct. Anal. 19(4), 1017–1028 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  13. Cheeger, J., Kleiner, B., Naor, A.: Compression bounds for Lipschitz maps from the Heisenberg group to L 1. arXiv:0910.2026 (2009)

  14. Cheeger, J., Kleiner, B., Naor, A.: A (log n)Ω(1) integrality gap for the sparsest cut SDP. In: Proceedings of 50th Annual IEEE Symposium on Foundations of Computer Science, FOCS, pp. 555–564 (2009)

  15. Dacunha-Castelle, D., Krivine, J.L.: Applications des ultraproduits à l’étude des espaces et des algèbres de Banach. Stud. Math. 41, 315–334 (1972)

    MATH  MathSciNet  Google Scholar 

  16. Eskin, A., Fisher, D., Whyte, K.: Quasi-isometries and rigidity of solvable groups. Pure Appl. Math. Q. 3(4, part 1), 927–947 (2007)

    MATH  MathSciNet  Google Scholar 

  17. Franchi, B., Serapioni, R., Serra Cassano, F.: Rectifiability and perimeter in the Heisenberg group. Math. Ann. 321(3), 479–531 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Franchi, B., Serapioni, R., Serra Cassano, F.: On the structure of finite perimeter sets in step 2 Carnot groups. J. Geom. Anal. 13(3), 421–466 (2003)

    MATH  MathSciNet  Google Scholar 

  19. Gromov, M.: Asymptotic invariants of infinite groups. In: Geometric Group Theory, Vol. 2, Sussex, 1991. London Math. Soc. Lecture Note Ser., vol. 182, pp. 1–295. Cambridge Univ. Press, Cambridge (1993)

    Google Scholar 

  20. Heinonen, J., Koskela, P.: From local to global in quasiconformal structures. Proc. Natl. Acad. Sci. USA 93, 554–556 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  21. Heinrich, S.: Ultraproducts in Banach space theory. J. Reine Angew. Math. 313, 72–104 (1980)

    MATH  MathSciNet  Google Scholar 

  22. Heinrich, S., Mankiewicz, P.: Applications of ultrapowers to the uniform and Lipschitz classification of Banach spaces. Stud. Math. 73(3), 225–251 (1982)

    MATH  MathSciNet  Google Scholar 

  23. Kakutani, S.: Mean ergodic theorem in abstract (L)-spaces. Proc. Imp. Acad., Tokyo 15, 121–123 (1939)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kirchheim, B.: Rectifiable metric spaces: local structure and regularity of the Hausdorff measure. Proc. Am. Math. Soc. 121(1), 113–123 (1994)

    MATH  MathSciNet  Google Scholar 

  25. Kleiner, B., Leeb, B.: Rigidity of quasi-isometries for symmetric spaces and Euclidean buildings. Inst. Hautes Études Sci. Publ. Math. 86, 115–197 (1998) 1997

    Article  Google Scholar 

  26. Lee, J., Naor, A.: L p metrics on the Heisenberg group and the Goemans-Linial conjecture. In: FOCS, pp. 99–108 (2006)

  27. Lee, James R., Raghavendra, Prasad: Coarse differentiation and multi-flows in planar graphs. Discrete Comput. Geom. 43(2), 346–362 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  28. Linial, N.: Finite metric-spaces—combinatorics, geometry and algorithms. In: Proceedings of the International Congress of Mathematicians, Vol. III, Beijing, 2002, pp. 573–586. Higher Ed. Press, Beijing (2002)

    Google Scholar 

  29. Linial, N., London, E., Rabinovich, Y.: The geometry of graphs and some of its algorithmic applications. Combinatorica 15(2), 215–245 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  30. Pansu, P.: Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un. Ann. Math. (2) 129(1), 1–60 (1989)

    Article  MathSciNet  Google Scholar 

  31. Pauls, S.: The large scale geometry of nilpotent Lie groups. Commun. Anal. Geom. 9(5), 951–982 (2001)

    MATH  MathSciNet  Google Scholar 

  32. Strichartz, R.S.: L p harmonic analysis and Radon transforms on the Heisenberg group. J. Funct. Anal. 96(2), 350–406 (1991)

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to Jeff Cheeger.

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Research supported in part by NSF grant DMS-0704404.

Research supported in part by NSF grant DMS-0805939.

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Cheeger, J., Kleiner, B. Metric differentiation, monotonicity and maps to L 1 . Invent. math. 182, 335–370 (2010). https://doi.org/10.1007/s00222-010-0264-9

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  • DOI: https://doi.org/10.1007/s00222-010-0264-9

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