Calculus of Variations and Geometric Measure Theory

P. Lahti - R. Jones - N. Shanmugalingam

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

created by shanmugal on 14 Aug 2019


Accepted Paper

Inserted: 14 aug 2019
Last Updated: 14 aug 2019

Journal: Indiana University Math. J.
Year: 2019


We generalize a result of Kelly~\cite{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 $\mathcal{H}^{Q-1}\vert_{\Sigma E}$ 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 $\Sigma E$, $f(E)$ 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.

Tags: GeMeThNES