Exceptional Sets for Quasiconformal Mappings in General Metric Spaces II

TL;DR

This paper investigates the conditions under which quasiconformal mappings in metric spaces can have their dilatation bounded outside small exceptional sets, establishing the sharpness of previous results related to Poincaré inequalities.

Contribution

It proves the optimality of allowing exceptional sets of codimension p in metric spaces with Poincaré inequalities for quasiconformal mappings.

Findings

Established the sharpness of the codimension p exceptionality condition.

Extended classical Euclidean results to general metric spaces.

Demonstrated the necessity of Poincaré inequalities for these properties.

Abstract

A homemorphism between domains in $\mathbb R^n$, $n\ge 2$ is quasiconformal, with its intricate analytic and geometric consequences, if the (pointwise) linear dilatation -- a purely metric quantity -- is uniformly bounded. Gehring proved that it will suffice to verify the uniform bound up to a set of measure zero as long as we can show that the dilatation is finite outside a subset of finite Hausdorff--$(n-1)$ measure. In short, we say that we can allow an exceptional codimension $1$ subset. In the metric setting, it has been proved, roughly speaking, that one can allow an exceptional codimension $p$ subset, $p \ge 1$, if the source space satisfies a $p$-Poincar\'e inequality. We prove, effectively, the sharpness of the latter claim.