Non-removability of the Sierpinski Gasket

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 ${\mathbb{R}}^2$ into some non-planar surface $S$, and then embedding this surface quasisymmetrically back into the plane by using the celebrated Bonk-Kleiner Theorem 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 $1\le p\le 2$, thus complementing and sharpening the results of the author's previous work arXiv:1706.07687.

arXiv comment61 pages, 9 figures

References in corpus (7)

• Quasisymmetric parametrizations of two-dimensional metric spheres
• Non-uniform hyperbolicity in complex dynamics
• Quasisymmetric parameterizations of two-dimensional metric planes
• Canonical parametrizations of metric discs
• Holomorphic Removability of Julia Sets
• Non-removability of Sierpinski carpets
• Non-removable sets for quasiconformal and locally biLipschitz mappings in R^3

Cited by in corpus (6)

• Uniqueness of the welding problem for SLE and Liouville quantum gravity
• Non-removability of Sierpinski carpets
• CNED sets: countably negligible for extremal distances
• Non-removability of Sierpinski spaces
• A removability theorem for Sobolev functions and detour sets
• David extension of circle homeomorphisms, welding, mating, and removability