Lipschitz maps and nets in Euclidean space. (English) Zbl 0941.37030

McMullen, C. T.

Geom. Funct. Anal. 8, No. 2, 304-314 (1998).

This paper deals with the following three questions:
1. Is there a bi-Lipschitz homeomorphism \(\varphi:\mathbb{R}^n\to\mathbb{R}^n\) such that the Jacobian determinant \(\text{det }D\varphi=f\), where \(f\) is a given real-valued function \(f\in L^\infty(\mathbb{R}^n)\) with \(\inf f(x)>0\)?
2. Is there a Lipschitz or quasiconformal vector field with \(\text{div }v=f\), where \(f\) is a given \(f\in L^\infty(\mathbb{R}^n)\)?
3. Given a separated net \(Y\subset \mathbb{R}^n\), is there a bi-Lipschitz map \(\varphi:Y\to\mathbb{Z}^n\)?
Note that in the case of \(n=1\) all three questions have a positive answer. The author shows that for \(n>1\) the answer to all three questions is negative. He also proves that all three questions have positive solutions if the Lipschitz condition is replaced by a Hölder condition.

Reviewer: Messoud Efendiev (Berlin)

MSC:

37F30	Quasiconformal methods and Teichmüller theory, etc. (dynamical systems) (MSC2010)
37F05	Dynamical systems involving relations and correspondences in one complex variable

Keywords:

bi-Lipschitz homeomorphism; Jacobian determinant; quasiconformal vector field; bi-Lipschitz map