A note on the injectivity radius of quasiconformal immersions (Q2815677)

In [Math. USSR, Sb. 3, 389--403 (1969; Zbl 0184.10801)], the author proved that a locally homeomorphic quasiconformal mapping \(f : \mathbb{R}^n \rightarrow \mathbb{R}^n\) is a homeomorphism provided that \(n \geq 3\). This result was conjectured by \textit{L. A. Lavrentieff} [C. R. (Dokl.) Acad. Sci. URSS, n. Ser. 20, 241--242 (1938; Zbl 0019.40306)] for \(n= 3\). In [\textit{O. Martio} et al., Ann. Acad. Sci. Fenn., Ser. A I 488, 31 p. (1971; Zbl 0223.30018)] it was shown that a locally homeomorphic \(K\)-quasiconformal mapping of the unit ball \(B(0,1)\) into \(\mathbb{R}^n\), \(n \geq 3\), is homeomorphic in the ball \(B(0,r)\) where \(r > 0\) depends only on \(n\) and \(K\). Here the author gives a short proof for the latter result based on a normal family argument and on a result of his own. This argument does not produce any explicit lower bounds for \(r\) as the proof in the aforementioned paper. Sharp estimates for \(r\) are not known at present. The author also discusses the situation in spaces other than the Euclidean spaces.