Typical Lipschitz images of rectifiable metric spaces

Abstract

This article studies typical 1-Lipschitz images of 𝑛-rectifiable metric spaces 𝐸 into R m \mathbb{R}^{m} for m ≥ n m\geq n . For example, if E ⊂ R k E\subset\mathbb{R}^{k} , we show that the Jacobian of such a typical 1-Lipschitz map equals 1 H n \mathcal{H}^{n} -almost everywhere and, if m > n 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 | ⋅ | \lvert\,{\cdot}\,\rvert on R m \mathbb{R}^{m} , we show that, in the space of 1-Lipschitz functions from ( [ − 1 , 1 ] n , | ⋅ | ∞ ) ([-1,1]^{n},\lvert\,{\cdot}\,\rvert_{\infty}) to ( R m , | ⋅ | ) (\mathbb{R}^{m},\lvert\,{\cdot}\,\rvert) , the H n \mathcal{H}^{n} -measure of a typical image is not bounded below by any Δ > 0 \Delta>0 .