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arXiv · 1510.05847

Geometric properties of $φ$-uniform domains

Peter Hästö·Riku Klén·Swadesh Kumar Sahoo·Matti Vuorinen
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Abstract

We consider proper subdomains $G$ of $\mathbb{R}^n$ and their images $G'=f(G)$ under quasiconformal mappings $f$ of $\mathbb{R}^n$. We compare the distance ratio metrics of $G$ and $G'$; as an application we show that $φ$-uniform domains are preserved under quasiconformal mappings of $\mathbb{R}^n$. A sufficient condition for $φ$-uniformity is obtained in terms of the quasi-symmetry condition. We give a geometric condition for uniformity: If $G\subset\mathbb{R}^n$ is $ϕ$-uniform and satisfies the twisted cone condition, then it is uniform. We also construct a planar $ϕ$-uniform domain whose complement is not $ψ$-uniform for any $ψ$.

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Peter Hästö, Riku Klén, Swadesh Kumar Sahoo, Matti Vuorinen. 2017-03-24. Geometric properties of $φ$-uniform domains. https://doi.org/10.1007/s41478-016-0011-8

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