On BLD-mappings with small distortion


We show that every \(L\)-BLD-mapping in a domain of \(\mathbb {R}^{n}\) is a local homeomorphism if \(L < \sqrt{2}\) or \(K_I(f) < 2\). These bounds are sharp as shown by a winding map.

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  1. 1.

    Drasin, D., Pankka, P.: Sharpness of Rickman’s Picard theorem in all dimensions. Acta Math. 214(2), 209–306 (2015)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Gehring, F.W.: Rings and quasiconformal mappings in space. Trans. Am. Math. Soc. 103, 353–393 (1962)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Gol’dšteĭn, V.M.: The behavior of mappings with bounded distortion when the distortion coefficient is close to one. Sibirsk. Mat. Ž. 12, 1250–1258 (1971)

    MathSciNet  Google Scholar 

  4. 4.

    Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics, vol. 152. Birkhäuser, Boston, MA (1999)

    Google Scholar 

  5. 5.

    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  6. 6.

    Heinonen, J., Kilpeläinen, T.: BLD-mappings in \(W^{2,2}\) are locally invertible. Math. Ann. 318(2), 391–396 (2000)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Hajłasz, P., Malekzadeh, S., Zimmerman, S.: Weak BLD mappings and Hausdorff measure. Nonlinear Anal. 177(1), 524–531 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Heinonen, J., Sullivan, D.: On the locally branched Euclidean metric gauge. Duke Math. J. 114(1), 15–41 (2002)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Iwaniec, T., Martin, G.: Geometric Function Theory and Non-linear Analysis. Oxford Mathematical Monographs. Oxford University Press, New York (2001)

    Google Scholar 

  10. 10.

    Kauranen, A., Luisto, R., Tengvall, V.: Mappings of finite distortion: compactness of the branch set. (to appear in J. Anal. Math.), arXiv:1709.08724v3

  11. 11.

    Le Donne, E., Pankka, P.: Closed BLD-elliptic manifolds have virtually Abelian fundamental groups. N. Y. J. Math. 20, 209–216 (2014)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Luisto, R.: Note on local-to-global properties of BLD-mappings. Proc. Am. Math. Soc. 144(2), 599–607 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Luisto, R.: A characterization of BLD-mappings between metric spaces. J. Geom. Anal. 27(3), 2081–2097 (2017)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Martio, O., Rickman, S., Väisälä, J.: Topological and metric properties of quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I(488), 31 (1971)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Martio, O., Väisälä, J.: Elliptic equations and maps of bounded length distortion. Math. Ann. 282(3), 423–443 (1988)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Rajala, K.: The local homeomorphism property of spatial quasiregular mappings with distortion close to one. Geom. Funct. Anal. 15(5), 1100–1127 (2005)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Reshetnyak, J.G.: Liouville’s conformal mapping theorem under minimal regularity hypotheses. Sibirsk. Mat. Ž. 8, 835–840 (1967)

    MathSciNet  Google Scholar 

  18. 18.

    Reshetnyak, Y.G.: Space mappings with bounded distortion, Translations of Mathematical Monographs, vol 73. American Mathematical Society, Providence, RI (1989). Translated from the Russian by H. H. McFaden

  19. 19.

    Rickman, S.: Quasiregular Mappings, vol. 26. Springer, Berlin (1993)

    Book  Google Scholar 

  20. 20.

    Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics, vol. 1319. Springer, Berlin (1988)

    Book  Google Scholar 

  21. 21.

    Zorič, V.A.: MA Lavrent’ev’s theorem on quasiconformal space maps. Mat. Sb. N.S. 74(116):417–433 (1967)

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Correspondence to Rami Luisto.

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Dedicated to Professor Pekka Koskela on his 59th birthday.

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A.K. acknowledges the support of Academy of Finland, Grant Number 322441

The research of V.T. was supported by the Academy of Finland, Project Number 308759.

R.L. was partially supported by the Academy of Finland (grant 288501 ‘Geometry of subRiemannian groups’) and by the European Research Council (ERC Starting Grant 713998 GeoMeG ‘Geometry of Metric Groups’)

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Kauranen, A., Luisto, R. & Tengvall, V. On BLD-mappings with small distortion. Complex Anal Synerg 7, 5 (2021). https://doi.org/10.1007/s40627-021-00067-y

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  • BLD-mappings
  • branch set
  • quasiregular mappings
  • local homeomorphism

Mathematics Subject Classification

  • 57M12
  • 30C65