Teichm\"uller's problem in space

TL;DR

This paper extends Teichmüller's classical results to higher-dimensional space, providing explicit bounds for quasiconformal homeomorphisms of R^n, and explores their stability and metric behavior.

Contribution

It offers a spatial analogue of Teichmüller's theorem with explicit bounds, utilizing distortion results and special functions, and analyzes quasihyperbolic metric behavior under quasiconformal maps.

Findings

Explicit bounds for quasiconformal maps in R^n

Stability analysis as dilatation tends to 1

Sharp results for quasihyperbolic metric behavior

Abstract

Quasiconformal homeomorphisms of the whole space Rn, onto itself normalized at one or two points are studied. In particular, the stability theory, the case when the maximal dilatation tends to 1, is in the focus. Our main result provides a spatial analogue of a classical result due to Teichm\"uller. Unlike Teichm\"uller's result, our bounds are explicit. Explicit bounds are based on two sharp well-known distortion results: the quasiconformal Schwarz lemma and the bound for linear dilatation. Moreover, Bernoulli type inequalities and asymptotically sharp bounds for special functions involving complete elliptic integrals are applied to simplify the computations. Finally, we discuss the behavior of the quasihyperbolic metric under quasiconformal maps and prove a sharp result for quasiconformal maps of R^n \ {0} onto itself.