Non-removability of the Sierpinski Gasket

TL;DR

This paper demonstrates that the Sierpiński gasket cannot be removed for quasiconformal maps and continuous Sobolev functions, using novel constructions and quasisymmetric embeddings, thereby resolving a longstanding question and extending previous results.

Contribution

It introduces a new technique for constructing exceptional homeomorphisms and proves non-removability of the Sierpiński gasket for quasiconformal and Sobolev functions, advancing understanding in geometric function theory.

Findings

Sierpiński gasket is non-removable for quasiconformal maps.

All homeomorphic copies of the gasket are non-removable for $W^{1,p}$ functions, $1\, ext{to}\,2$.

New construction technique for exceptional homeomorphisms.

Abstract

We prove that the Sierpi\'nski 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 Sierpi\'nski gasket are non-removable for continuous Sobolev functions of the class $W^{1,p}$ for $1\leq p\leq 2$, thus complementing and sharpening the results of the author's previous work arXiv:1706.07687.