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Non-removability of the Sierpiński gasket

  • Dimitrios NtalampekosEmail author
Article
  • 111 Downloads

Abstract

We prove that the Sierpiński gasket is non-removable for quasiconformal maps, thus answering a question of Bishop (NSF Research Proposal, 2015. http://www.math.stonybrook.edu/~bishop/vita/nsf15.pdf). The proof involves a new technique of constructing an exceptional homeomorphism from \(\mathbb {R}^2\) into some non-planar surface S, and then embedding this surface quasisymmetrically back into the plane by using the celebrated Bonk–Kleiner Theorem (Bonk and Kleiner in Invent Math 150(1):127–183, 2002). We also prove that all homeomorphic copies of the Sierpiński gasket are non-removable for continuous Sobolev functions of the class \(W^{1,p}\) for \(1\le p\le 2\), thus complementing and sharpening the results of the author’s previous work (Ntalampekos in A removability theorem for Sobolev functions and detour sets. arXiv:1706.07687).

Mathematics Subject Classification

Primary 30C62 Secondary 46E35 30L10 51F99 

Notes

Acknowledgements

I am grateful to my advisor at UCLA, Mario Bonk, not only for the numerous conversations and useful comments during this project, but also for introducing me to the world of analysis on metric spaces and constantly keeping me motivated to learn mathematics and work on fascinating problems. I also thank Huy Tran for bringing the problem of (non)-removability of the gasket to my attention, Malik Younsi for several motivating conversations, Pekka Koskela for his suggestions regarding the proof of Theorem 1.3, and Guy C. David for explaining the different notions of convergence of metric spaces that appear in the literature. Moreover I would like to thank Matthew Romney, Raanan Schul, Jang-Mei Wu, Malik Younsi, and the anonymous referee for their comments and corrections.

References

  1. 1.
    Ambrosio, L., Tilli, P.: Topics on Analysis in Metric Spaces. Oxford University Press, Oxford (2004)zbMATHGoogle Scholar
  2. 2.
    Astala, K., Iwaniec, T., Martin, G.: Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane. Princeton University Press, Princeton (2009)zbMATHGoogle Scholar
  3. 3.
    Besicovitch, A.S.: On sufficient conditions for a function to be analytic, and on behaviour of analytic functions in the neighbourhood of non-isolated singular points. Proc. Lond. Math. Soc. (2) 32(1), 1–9 (1931)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bishop, C.: Some homeomorphisms of the sphere conformal off a curve. Ann. Acad. Sci. Fenn. Ser. A I Math. 19(2), 323–338 (1994)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Bishop, C.: Non-removable sets for quasiconformal and locally biLipschitz mappings in \(\mathbb{R}^3\). Stony Brook IMS (1998). Preprint. http://www.math.stonybrook.edu/cgi-bin/preprint.pl?ims98-6. Accessed 10 Dec 2017
  6. 6.
    Bishop, C.: NSF Research Proposal (2015). http://www.math.stonybrook.edu/~bishop/vita/nsf15.pdf. Accessed 10 Dec 2017
  7. 7.
    Bonk, M., Kleiner, B.: Quasisymmetric parametrizations of two-dimensional metric spheres. Invent. Math. 150(1), 127–183 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bonk, M., Meyer, D.: Expanding Thurston Maps, Mathematical Surveys and Monographs, vol. 225. American Mathematical Society, Providence (2017)CrossRefzbMATHGoogle Scholar
  9. 9.
    Burago, D., Burago, Y., Ivanov, S.: A Course in Metric Geometry, Graduate Studies in Mathematics, vol. 33. American Mathematical Society, Providence (2001)zbMATHGoogle Scholar
  10. 10.
    David, G., Semmes, S.: Fractured Fractals and Broken Dreams. Oxford University Press, New York (1997)zbMATHGoogle Scholar
  11. 11.
    Federer, H., Ziemer, W.P.: The Lebesgue set of a function whose distribution derivatives are \(p\)-th power summable. Indiana Univ. Math. J. 22, 139–158 (1972/73)Google Scholar
  12. 12.
    Folland, G.B.: Real Analysis, 2nd edn. Wiley, New York (1999)zbMATHGoogle Scholar
  13. 13.
    Gehring, F.W.: The definitions and exceptional sets for quasiconformal mappings. Ann. Acad. Sci. Fenn. Ser. A I(281), 0–28 (1960)Google Scholar
  14. 14.
    Graczyk, J., Smirnov, S.: Non-uniform hyperbolicity in complex dynamics. Invent. Math. 175(2), 335–415 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Heinonen, J.: Lectures on Analysis on Metric Spaces. Springer, New York (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181(1), 1–61 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    He, Z.-X., Schramm, O.: Rigidity of circle domains whose boundary has \(\sigma \)-finite linear measure. Invent. Math. 115(2), 297–310 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Jones, P.: On Removable Sets for Sobolev Spaces in the Plane, Essays on Fourier Analysis in Honor of Elias M. Stein (Princeton, NJ, 1991). Princeton Mathematics Series, vol. 42, pp. 250–276. Princeton University Press, Princeton (1995)Google Scholar
  19. 19.
    Jones, P., Smirnov, S.: Removability theorems for Sobolev functions and quasiconformal maps. Ark. Mat. 38(2), 263–279 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kahn, J.: Holomorphic Removability of Julia Sets. Preprint. arXiv:math/9812164
  21. 21.
    Kaufman, R.: Fourier-Stieltjes coefficients and continuation of functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 9, 27–31 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Kaufman, R., Wu, J.-M.: On removable sets for quasiconformal mappings. Ark. Mat. 34(1), 141–158 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kinneberg, K.: Lower bounds for codimension-1 measure in metric manifolds. Rev. Mat. Iberoam. 34(3), 1103–1118 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Keith, S., Laakso, T.: Conformal assouad dimension and modulus. Geom. Funct. Anal. 14(6), 1278–1321 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Koskela, P., Rajala, T., Zhang, Y.: A density problem for Sobolev functions on Gromov hyperbolic domains. Nonlinear Anal. 154, 189–209 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Koskela, P., Nieminen, T.: Quasiconformal removability and the quasihyperbolic metric. Indiana Univ. Math. J. 54(1), 143–151 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Lytchak, A., Wenger, S.: Canonical parametrizations of metric discs. Preprint. arXiv:1701.06346
  28. 28.
    Merenkov, S., Wildrick, K.: Quasisymmetric Koebe uniformization. Rev. Mat. Iberoam. 29(3), 859–909 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Moore, R.L.: Foundations of Point Set Theory, American Mathematical Society Colloquium Publications. American Mathematical Society, Providence (1962)Google Scholar
  30. 30.
    Ntalampekos, D.: A removability theorem for Sobolev functions and detour sets. Preprint. arXiv:1706.07687
  31. 31.
    Ntalampekos, D.: Non-removability of Sierpinski carpets. Preprint. arXiv:1809.05605
  32. 32.
    Ntalampekos, D., Younsi, M.: Rigidity theorems for circle domains. Preprint. arXiv:1809.05573
  33. 33.
    Rajala, K.: Uniformization of two-dimensional metric surfaces. Invent. Math. 207(3), 1301–1375 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sheffield, S.: Conformal weldings on random surfaces: SLE and the quantum gravity zipper. Ann. Probab. 44(5), 3474–3545 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Smith, W., Stegenga, D.: Hölder domains and Poincaré domains. Trans. Am. Math. Soc. 319(1), 67–100 (1990)zbMATHGoogle Scholar
  36. 36.
    Väisälä, J.: Lectures on \(n\)-Dimensional Quasiconformal Mappings, Lecture Notes in Mathematics, vol. 229. Springer, Berlin (1971)CrossRefzbMATHGoogle Scholar
  37. 37.
    Whyburn, G.T.: Topological characterization of the Sierpiński curve. Fund. Math. 45, 320–324 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Wildrick, K.: Quasisymmetric parametrizations of two-dimensional metric planes. Proc. Lond. Math. Soc. (3) 97(3), 783–812 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Wu, J.-M.: Removability of sets for quasiconformal mappings and Sobolev spaces. Complex Var. Theory Appl. 37(1–4), 491–506 (1998)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Younsi, M.: On removable sets for holomorphic functions. EMS Surv. Math. Sci. 2(2), 219–254 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Younsi, M.: Removability, rigidity of circle domains and Koebe’s conjecture. Adv. Math. 303, 1300–1318 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute for Mathematical SciencesStony Brook UniversityStony BrookUSA

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