Typical Lipschitz images of rectifiable metric spaces

Abstract

This article studies typical 1-Lipschitz images of $n$-rectifiable metric spaces $E$ into ${\mathbb{R}}^m$ for $m\ge n$. For example, if $E\subset {\mathbb{R}}^k$, we show that the Jacobian of such a typical 1-Lipschitz map equals 1 ${\mathcal{H}}^n$-almost everywhere and, if $m>n$, preserves the Hausdorff measure of $E$. In general, we provide sufficient conditions, in terms of the tangent norms of $E$, for when a typical 1-Lipschitz map preserves the Hausdorff measure of $E$, up to some constant multiple. Almost optimal results for strongly $n$-rectifiable metric spaces are obtained. On the other hand, for any norm $|\cdot |$ on ${\mathbb{R}}^m$, we show that, in the space of 1-Lipschitz functions from $([-1,1{]}^n,|\cdot {|}_{\infty })$ to $({\mathbb{R}}^m,|\cdot |)$, the ${\mathcal{H}}^n$-measure of a typical image is not bounded below by any $\Delta >0$.

References in corpus (4)

• Structure Theory of Metric-Measure Spaces with Lower Ricci Curvature Bounds
• On the volume measure of non-smooth spaces with Ricci curvature bounded below
• Purely unrectifiable metric spaces and perturbations of Lipschitz functions
• Characterising rectifiable metric spaces using tangent spaces