Modulus of families of sets of finite perimeter and quasiconformal maps between metric spaces of globally $Q$-bounded geometry

TL;DR

This paper extends the understanding of how quasiconformal maps between metric spaces with certain regularity properties preserve the modulus of measures associated with sets of finite perimeter, generalizing previous Euclidean results.

Contribution

It generalizes Kelly's result to Ahlfors regular metric spaces supporting a Poincaré inequality, showing quasi-preservation of measure moduli under quasiconformal maps.

Findings

Quasiconformal maps quasi-preserve the modulus of measures related to finite perimeter sets.

Under certain conditions, images of finite perimeter sets remain of finite perimeter.

Results are more general than previous Euclidean cases, applicable to a broader class of metric spaces.

Abstract

We generalize a result of J. C. Kelly to the setting of Ahlfors $Q$-regular metric measure spaces supporting a $1$-Poincar\'e inequality. It is shown that if $X$ and $Y$ are two Ahlfors $Q$-regular spaces supporting a $1$-Poincar\'e inequality and $f:X\to Y$ is a quasiconformal mapping, then the $Q/(Q-1)$-modulus of the collection of measures $\mathcal{H}^{Q-1}\lfloor_{\Sigma E}$ corresponding to any collection of sets $E\subset X$ of finite perimeter is quasi-preserved by $f$. We also show that for $Q/(Q-1)$-modulus almost every $\Sigma E$, if the image surface $\Sigma f(E)$ does not see the singular set of $f$ as a large set, then $f(E)$ is also of finite perimeter. Even in the standard Euclidean setting our results are more general than that of Kelly, and hence are new even in there.