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arXiv · 2106.06177

Existence of quasiconformal maps with maximal stretching on any given countable set

Rosemarie Bongers·James T. Gill
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Abstract

Quasiconformal maps are homeomorphisms with useful local distortion inequalities; infinitesimally, they map balls to ellipsoids with bounded eccentricity. This leads to a number of useful regularity properties, including quantitative H\"older continuity estimates; on the other hand, one can use the radial stretches to characterize the extremizers for H\"older continuity. In this work, given any bounded countable set in $\mathbb{R}^d$, we will construct an example of a $K$-quasiconformal map which exhibits the maximum stretching at each point of the set. This will provide an example of a quasiconformal map that exhibits the worst-case regularity on a surprisingly large set, and generalizes constructions from the planar setting into $\mathbb{R}^d$.

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BibTeXRIS

Rosemarie Bongers, James T. Gill. 2021-06-11. Existence of quasiconformal maps with maximal stretching on any given countable set. https://arxiv.org/abs/2106.06177

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