Mappings of finite distortion from generalized manifolds
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- by Ville Kirsilä
- Conform. Geom. Dyn. 18 (2014), 229-262
- DOI: https://doi.org/10.1090/S1088-4173-2014-00272-0
- Published electronically: November 17, 2014
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Abstract:
We give a definition for mappings of finite distortion from a generalized manifold with controlled geometry to a Euclidean space. We prove that the basic properties of mappings of finite distortion are valid in this context. In particular, we show that under the same assumptions as in the Euclidean case, mappings of finite distortion are open and discrete.References
- Daniel Aalto and Juha Kinnunen, The discrete maximal operator in metric spaces, J. Anal. Math. 111 (2010), 369–390. MR 2747071, DOI 10.1007/s11854-010-0022-3
- J. M. Ball, Global invertibility of Sobolev functions and the interpenetration of matter, Proc. Roy. Soc. Edinburgh Sect. A 88 (1981), no. 3-4, 315–328. MR 616782, DOI 10.1017/S030821050002014X
- Zoltán M. Balogh, Kevin Rogovin, and Thomas Zürcher, The Stepanov differentiability theorem in metric measure spaces, J. Geom. Anal. 14 (2004), no. 3, 405–422. MR 2077159, DOI 10.1007/BF02922098
- J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428–517. MR 1708448, DOI 10.1007/s000390050094
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Piotr Hajłasz and Pekka Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), no. 688, x+101. MR 1683160, DOI 10.1090/memo/0688
- Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
- Juha Heinonen and Stephen Keith, Flat forms, bi-Lipschitz parameterizations, and smoothability of manifolds, Publ. Math. Inst. Hautes Études Sci. 113 (2011), 1–37. MR 2805596, DOI 10.1007/s10240-011-0032-4
- Juha Heinonen and Pekka Koskela, Quasiconformal maps in metric spaces with controlled geometry, Acta Math. 181 (1998), no. 1, 1–61. MR 1654771, DOI 10.1007/BF02392747
- Juha Heinonen, Pekka Koskela, Nageswari Shanmugalingam, and Jeremy T. Tyson, Sobolev classes of Banach space-valued functions and quasiconformal mappings, J. Anal. Math. 85 (2001), 87–139. MR 1869604, DOI 10.1007/BF02788076
- J. Heinonen and P. Koskela, N. Shanmugalingam, J. Tyson, Sobolev Spaces on metric measure spaces: an approach based on upper gradients, to appear.
- Juha Heinonen and Seppo Rickman, Geometric branched covers between generalized manifolds, Duke Math. J. 113 (2002), no. 3, 465–529. MR 1909607, DOI 10.1215/S0012-7094-02-11333-7
- Juha Heinonen and Dennis Sullivan, On the locally branched Euclidean metric gauge, Duke Math. J. 114 (2002), no. 1, 15–41. MR 1915034, DOI 10.1215/S0012-7094-02-11412-4
- Stanislav Hencl and Kai Rajala, Optimal assumptions for discreteness, Arch. Ration. Mech. Anal. 207 (2013), no. 3, 775–783. MR 3017286, DOI 10.1007/s00205-012-0574-8
- Tadeusz Iwaniec and Vladimír Šverák, On mappings with integrable dilatation, Proc. Amer. Math. Soc. 118 (1993), no. 1, 181–188. MR 1160301, DOI 10.1090/S0002-9939-1993-1160301-5
- Pekka Koskela and Paul MacManus, Quasiconformal mappings and Sobolev spaces, Studia Math. 131 (1998), no. 1, 1–17. MR 1628655
- J. Maly, A simple proof of the Stepanov theorem on differentiability almost everywhere, Exposition. Math. 17 (1999), no. 1, 59–61. MR 1687460
- Juan J. Manfredi and Enrique Villamor, Mappings with integrable dilatation in higher dimensions, Bull. Amer. Math. Soc. (N.S.) 32 (1995), no. 2, 235–240. MR 1313107, DOI 10.1090/S0273-0979-1995-00583-5
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- F. Morgan, Geometric measure theory A beginner’s guide, Academic Press.
- Jani Onninen and Xiao Zhong, Mappings of finite distortion: a new proof for discreteness and openness, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), no. 5, 1097–1102. MR 2477453, DOI 10.1017/S0308210506000825
- Kai Rajala, Remarks on the Iwaniec-Šverák conjecture, Indiana Univ. Math. J. 59 (2010), no. 6, 2027–2039. MR 2919747, DOI 10.1512/iumj.2010.59.3946
- Seppo Rickman, Quasiregular mappings, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 26, Springer-Verlag, Berlin, 1993. MR 1238941, DOI 10.1007/978-3-642-78201-5
- S. Semmes, Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities, Selecta Math. (N.S.) 2 (1996), no. 2, 155–295. MR 1414889, DOI 10.1007/BF01587936
- Nageswari Shanmugalingam, Newtonian spaces: an extension of Sobolev spaces to metric measure spaces, Rev. Mat. Iberoamericana 16 (2000), no. 2, 243–279. MR 1809341, DOI 10.4171/RMI/275
- C. J. Titus and G. S. Young, The extension of interiority, with some applications, Trans. Amer. Math. Soc. 103 (1962), 329–340. MR 137103, DOI 10.1090/S0002-9947-1962-0137103-6
- Jussi Väisälä, Minimal mappings in euclidean spaces, Ann. Acad. Sci. Fenn. Ser. A I 366 (1965), 22. MR 0178454
Bibliographic Information
- Ville Kirsilä
- Affiliation: Department of Mathematics and Statistics (P.O. Box 35 (MaD)), FI-40014 University of Jyväskylä, Finland
- Email: ville.kirsila@jyu.fi
- Received by editor(s): June 26, 2014
- Received by editor(s) in revised form: October 13, 2014
- Published electronically: November 17, 2014
- Additional Notes: The author was financially supported by the Finnish National Doctoral Programme in Mathematics and its Applications and by the Academy of Finland, project 257482.
- © Copyright 2014 American Mathematical Society
- Journal: Conform. Geom. Dyn. 18 (2014), 229-262
- MSC (2010): Primary 30C65; Secondary 30L10
- DOI: https://doi.org/10.1090/S1088-4173-2014-00272-0
- MathSciNet review: 3278159