Quasisymmetric embeddings of products of cells into the Euclidean space. (English) Zbl 0807.30011

Alestalo, Pekka; Väisälä, Jussi

Ann. Acad. Sci. Fenn., Ser. A I, Math. 19, No. 2, 375-392 (1994).

We shall mainly consider the following question. Suppose that \(A\) is a \(p\)-cell, \(B\) is a \(q\)-cell and there exists a quasisymmetric embedding \(f\colon A\times B\to \mathbb{R}^{p+q}\). What can we then say about the metric properties of \(A\) and \(B\)? It turns out that, for example, the cell \(A\) locally satisfies a \((p-1)\)-dimensional bounded turning condition. We also study the measures of \(A\) and \(B\) and show that if \(B\) is a quasiconvex arc, this implies certain conditions for the measure of \(A\).

Reviewer: P.Alestalo (Helsinki)

MSC:

30C65

Quasiconformal mappings in \(\mathbb{R}^n\), other generalizations

Keywords:

quasisymmetric map; bounded turning; quasiconvex arc