Non-removability of the Sierpinski Gasket

2018/04/30 by Dimitrios Ntalampekos
Mathematics · #math.MG #math.CA #math.CV #msc:30C62 #msc:46E35 #msc:30L10 #msc:51F99

paper · pdf · doi:10.1007/s00222-018-00852-3

published as Invent. Math. 216 (2019), no. 2, 519-595 · 61 pages, 9 figures

arxiv created 2018/12/13 · arxiv updated 2019/06/10

Abstract

We prove that the Sierpiński gasket is non-removable for quasiconformal maps, thus answering a question of Bishop. The proof involves a new technique of constructing an exceptional homeomorphism from \mathbb R² into some non-planar surface S, and then embedding this surface quasisymmetrically back into the plane by using the celebrated Bonk-Kleiner Theorem arXiv:math/0107171. 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≤ p≤ 2, thus complementing and sharpening the results of the author's previous work arXiv:1706.07687.

Citations

• Quasisymmetric parametrizations of two-dimensional metric spheres

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• Uniformization of cofat domains on metric two-spheres