From local to global in quasiconformal structures. (English) Zbl 0842.30016

Heinonen, Juha; Koskela, Pekka

Proc. Natl. Acad. Sci. USA 93, No. 2, 554-556 (1996).

Let \(d\) and \(d'\) denote two metrics on a space \(X\). Does a \(d'\)-metric ball \(B'(x, s)\) look round when viewed in terms of the \(d\) metric? One can quantify the answer in the following terms: let \(t\) be the largest radius such that the \(d\)-metric ball \(B(x, t)\) is contained in \(B'(x, s)\) and let \(u\) be the smallest radius such that the \(d\)-metric ball \(B(x, u)\) contains \(B'(x, s)\); then the ratio \(H= u/t\) is a number in the range \(1\leq H\leq \infty\) which measures the \(d\)-distortion of the \(d'\)-ball \(B'(x, s)\). The closer \(H\) is to 1, the nearer \(B'(x, s)\) is to \(d\)-round.
The authors say that \(d'\) is locally quasiconformally related to \(d\) if there is an \(H< \infty\) such that all sufficiently small \(d'\)-balls in \(X\) are pinched between \(d\)-balls as above with ratio of radii \(\leq H\).
If the same is true of arbitrary \(d'\)-balls, then they say that \(d'\) is globally quasiconformally related to \(d\).
This paper identifies conditions on spaces and metrics sufficient to imply that a local quasiconformal relationship implies a global quasiconformal relationship. Their result vastly extends the class of spaces where the result is known. The condition is described in terms of a Poincaré-type inequality.

Reviewer: J.W.Cannon (Provo)

MSC:

30C65

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

Keywords:

quasisymmetric mappings