Abstract
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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References
Drasin, D., Pankka, P.: Sharpness of Rickman’s Picard theorem in all dimensions. Acta Math. 214(2), 209–306 (2015)
Gehring, F.W.: Rings and quasiconformal mappings in space. Trans. Am. Math. Soc. 103, 353–393 (1962)
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)
Gromov, M.: Metric structures for Riemannian and non-Riemannian spaces. Progress in Mathematics, vol. 152. Birkhäuser, Boston, MA (1999)
Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)
Heinonen, J., Kilpeläinen, T.: BLD-mappings in \(W^{2,2}\) are locally invertible. Math. Ann. 318(2), 391–396 (2000)
Hajłasz, P., Malekzadeh, S., Zimmerman, S.: Weak BLD mappings and Hausdorff measure. Nonlinear Anal. 177(1), 524–531 (2018)
Heinonen, J., Sullivan, D.: On the locally branched Euclidean metric gauge. Duke Math. J. 114(1), 15–41 (2002)
Iwaniec, T., Martin, G.: Geometric Function Theory and Non-linear Analysis. Oxford Mathematical Monographs. Oxford University Press, New York (2001)
Kauranen, A., Luisto, R., Tengvall, V.: Mappings of finite distortion: compactness of the branch set. (to appear in J. Anal. Math.), arXiv:1709.08724v3
Le Donne, E., Pankka, P.: Closed BLD-elliptic manifolds have virtually Abelian fundamental groups. N. Y. J. Math. 20, 209–216 (2014)
Luisto, R.: Note on local-to-global properties of BLD-mappings. Proc. Am. Math. Soc. 144(2), 599–607 (2016)
Luisto, R.: A characterization of BLD-mappings between metric spaces. J. Geom. Anal. 27(3), 2081–2097 (2017)
Martio, O., Rickman, S., Väisälä, J.: Topological and metric properties of quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I(488), 31 (1971)
Martio, O., Väisälä, J.: Elliptic equations and maps of bounded length distortion. Math. Ann. 282(3), 423–443 (1988)
Rajala, K.: The local homeomorphism property of spatial quasiregular mappings with distortion close to one. Geom. Funct. Anal. 15(5), 1100–1127 (2005)
Reshetnyak, J.G.: Liouville’s conformal mapping theorem under minimal regularity hypotheses. Sibirsk. Mat. Ž. 8, 835–840 (1967)
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
Rickman, S.: Quasiregular Mappings, vol. 26. Springer, Berlin (1993)
Vuorinen, M.: Conformal Geometry and Quasiregular Mappings. Lecture Notes in Mathematics, vol. 1319. Springer, Berlin (1988)
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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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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DOI: https://doi.org/10.1007/s40627-021-00067-y