Quasiconformality of the injective mappings transforming spheres to quasispheres (Q511324)

In 1937 C. Carathéodory proved that an injective mapping from a domain \( D \subset \overline{\mathbb{C}}\) into \(\overline{\mathbb{C}}\) is a Möbius mapping if the image of each (generalized) circle in \(D\) belongs to the same category. No continuity was assumed. The corresponding theorems in \(\mathbb{R}^n\) were obtained in [\textit{E. Guyou}, C. R. Acad. Sci., Paris 128, 1197--1201 (1899; JFM 30.0839.02)] and [\textit{A. F. Beardon} and \textit{D. Minda}, Proc. Am. Math. Soc. 130, No. 4, 987--998 (2002; Zbl 0988.30005)]. The author proves a quasiconformal version of these results: Suppose that \(f\) is a locally injective mapping from a domain \(D \subset \overline{\mathbb{R}}^n\) into \(\overline{\mathbb{R}}^n\) and each point \(x \in D\) has a neighborhuud \(U\) such that the image of each sphere in \(U\) is a \(K\)-quasisphere, i.e., the image of a sphere under a global \(K\)-quasiconformal mapping of \(\overline{\mathbb{R}}^n\). Then \(f\) is a locally homeomorphic \(K'\)-quasiconformal mapping with \((K')^{1/(n-1)} \leq \exp\big(6(K+1)^2 (K-1)^{1/2}\big)\). The proof consists of two parts. First, assuming continuity, the author uses a quasimöbius mapping argument [\textit{J. Väisälä}, J. Anal. Math. 44, 218--234 (1985; Zbl 0593.30022)] to prove the result. The continuity of \(f\) requires several steps based on separation arguments.