The Journal of Geometric Analysis

, Volume 27, Issue 3, pp 2081–2097 | Cite as

A Characterization of BLD-Mappings Between Metric Spaces

  • Rami Luisto


We prove a characterization theorem for BLD-mappings between complete locally compact path-metric spaces. As a corollary, we obtain a sharp limit theorem for BLD-mappings.


BLD BLD-mappings Metric geometry Path-metric spaces Gromov-Hausdorff convergence Lipschitz quotient mappings Branched covers 

Mathematics Subject Classification

30L10 30C65 57M12 



The author would like to thank, once again, his advisor Pekka Pankka for introducing him to the world of BLD geometry. The ideas presented in this manuscript have been incubating for a long time and have been worked on during several wonderful mathematical events, but the author would like to mention the School in Geometric Analysis, Geometric analysis on Riemannian and singular metric spaces at Lake Como School of Advanced Studies in the fall of 2013 and the Research Term on Analysis and Geometry in Metric Spaces at ICMAT in the spring of 2015 as especially fruitful events concerning the current work. The author is especially grateful for all the interesting discussions with fellow mathematicians at these, and other, events. The contents of this paper were improved further by the author’s discussions with Piotr Hajłasz and his students while visiting University of Pittsburgh in 2016. Finally, the thoroughness and comments of the anonymous referee are gratefully acknowledged and have improved the readability of the manuscript. The author was supported by the Väisälä foundation.


  1. 1.
    Bates, S., Johnson, W.B., Lindenstrauss, J., Preiss, D., Schechtman, G.: Affine approximation of Lipschitz functions and nonlinear quotients. Geom. Funct. Anal. 9(6), 1092–1127 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Csörnyei, M.: Can one squash the space into the plane without squashing? Geom. Funct. Anal. 11(5), 933–952 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    David, G.C.: Tangents and rectifiability of Ahlfors regular Lipschitz differentiability spaces. Geom. Funct. Anal. 25(2), 553–579 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    David, G., Semmes, S.: Fractured Fractals and Broken Dreams. Oxford Lecture Series in Mathematics and its Applications, Vol 7. The Clarendon Press, Oxford University Press, New York (1997). Self-similar geometry through metric and measureGoogle Scholar
  5. 5.
    Floyd, E.E.: Some characterizations of interior maps. Ann. Math. 2(51), 571–575 (1950)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gromov, M.: Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, Vol 152. Birkhäuser Boston Inc., Boston (1999). Based on the 1981 French original [MR0682063 (85e:53051)], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael BatesGoogle Scholar
  7. 7.
    Guo, C.-Y., Williams, M.: Porosity of the Branch set of Discrete Open Mappings with Controlled Linear Dilatation (preprint)Google Scholar
  8. 8.
    Heinonen, J., Keith, S.: Flat forms, bi-Lipschitz parameterizations, and smoothability of manifolds. Publ. Math. Inst. Hautes Études Sci. 113, 1–37 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hajłasz, P., Malekzadeh, S.: A new characterization of the mappings of bounded length distortion. Int. Math. Res. Not. (2015)Google Scholar
  10. 10.
    Hajłasz, P., Malekzadeh, S.: On conditions for unrectifiability of a metric space. Anal. Geom. Metr. Spaces 3, 1–14 (2015)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Heinonen, J., Rickman, S.: Geometric branched covers between generalized manifolds. Duke Math. J. 113(3), 465–529 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kapovich, M.: Hyperbolic Manifolds and Discrete Groups. Modern Birkhäuser Classics. Birkhäuser Boston, Inc., Boston. Reprint of the 2001 edition (2009)Google Scholar
  13. 13.
    Kleiner, B., MacKay, J.: Differentiable structures on metric measure spaces: a primer. (preprint)Google Scholar
  14. 14.
    Kaczor, W.J., Nowak, M.T.: Problems in Mathematical Analysis. II, Student Mathematical Library, Vol. 12. American Mathematical Society, Providence (2001). Continuity and differentiation, Translated from the 1998 Polish original, revised and augmented by the authorsGoogle Scholar
  15. 15.
    LeDonne, E., Pankka, P.: Closed BLD-elliptic manifolds have virtually Abelian fundamental groups. N. Y. J. Math. 20, 209–216 (2014)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Luisto, R.: Note on local-to-global properties of BLD-mappings. Proc. Amer. Math. Soc. 144(2), 599–607 (2016)Google Scholar
  17. 17.
    Lytchak, A.: Open map theorem for metric spaces. Algebra i Anal. 17(3), 139–159 (2005)MathSciNetGoogle Scholar
  18. 18.
    Martio, O., Väisälä, J.: Elliptic equations and maps of bounded length distortion. Math. Ann. 282(3), 423–443 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Onninen, J., Rajala, K.: Quasiregular mappings to generalized manifolds. J. Anal. Math. 109, 33–79 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Rickman, S.: Path lifting for discrete open mappings. Duke Math. J. 40, 187–191 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rickman, S.: Quasiregular Mappings. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], Vol 26. Springer, Berlin, (1993)Google Scholar
  22. 22.
    Väisälä, J.: Discrete open mappings on manifolds. Ann. Acad. Sci. Fenn. Ser. A I No 392, 10 (1966)MathSciNetzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2016

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of HelsinkiHelsinkiFinland

Personalised recommendations