Estimates of stability, distortion theorems, and topological properties of quasiregular mappings (Q1316857)

Let \({\mathcal F}_ n\) be the class of all quasiregular maps \(f:R^ n \to \overline {R}^ n\), \(n \geq 3\), with the following properties: (1) \(f(0) = 0\), \(f(\infty) = \infty\), (2) in the unit ball \(B^ n\), for all \(x \in B^ n \setminus \{0\}\), \(f(x) \neq f(0)\), (3) \(B^ n \subset f(B^ n)\) and \(\min \{| f(x) | : | x | = 1\} = 1\). Further, for \(K \geq 1\), let \({\mathcal F}_ n(K)\) be the class of all those maps in \({\mathcal F}_ n\) whose linear dilatations are a.e. bounded by \(K\) and let \[ \mu_ n (K - 1) = \sup_{f \in {\mathcal F}_ n (K)} \inf_ A \| Af - id \|_{C(\overline {B}^ n)} \] where the infimum is taken over all orthogonal maps \(A:R^ n \to R^ n\) and \(id(x) = x\). The author shows that the extremal quantity \(\mu_ n\) is related in a simple way to several extremal problems and uses this quantity to prove the following theorem: If \(f:R^ n \to R^ n\) is quasiregular and \(\mu_ n (K(f) - 1) < \sqrt 2\), then \(f\) is injective. For \(n=2\) the author investigates some of the constants associated with the distortion properties of quasiregular maps. Complete elliptic integrals are used here.