Non-removability of the Sierpiński gasket

  • Dimitrios NtalampekosEmail author


We prove that the Sierpiński gasket is non-removable for quasiconformal maps, thus answering a question of Bishop (NSF Research Proposal, 2015. 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 



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.


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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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