Reifenberg flat metric spaces, snowballs, and embeddings
The main result here is to construct snowflake-like embeddings \(\varphi\) of flat enough metric spaces \(Z\) of dimension \(n\) into \(\mathbb{R}^{n+1}\) such that: \(C^{-1}\text{dist}(x,y)f(\text{dist}(x,y))\leq|\varphi(x)-\varphi(y)|C \text{dist}(x,y) f(\text{dist} (x,y))\), \(\forall x,y\in Z\), where \(f:(0,\infty)\to[1,\infty)\) is a given sufficiently regular function. When \(Z=\mathbb{R}^n\), then the mappings \(\varphi:\mathbb{R}^n\to\mathbb{R}^{n+1}\) satisfying the above inequalities: 1) can be extended to a quasiconformal mapping \(\widetilde\varphi\) of \(\mathbb{R}^{n+1}\), which is equal to the identity far from \(\mathbb{R}^n\); 2) are constructed here for the first time for \(f(r)=\max\{1,r^{-a}\}\) with some sufficiently small \(a>0\), and this construction provides an answer to Question 17 [\textit{J. Heinonen}, and \textit{S. Semmes}, Conform. Geom. Dyn. 1, 1-12 (1997; Zbl 0885.00006)]; 3) were constructed in [\textit{C. Bishop}, Proc. Am. Math. Soc. 127, 2035-2040 (1999; Zbl 0920.30016)] for functions \(f\) that grow very slowly to \(\infty\). The present results are also related to [\textit{J. Cheeger} and \textit{T. H. Colding}, J. Differ. Geom. 46, 406-480 (1997; Zbl 0902.53034)] on the parametrization of some metric spaces.
scientific article; zbMATH DE number 1395781