Stability property of Möbius mappings

Abstract:

Let $F$ be an arbitrary class of continuous mappings acting and ranging on domains in ${{\mathbf {R}}^n}$ which is invariant under similarity transformations of ${{\mathbf {R}}^n}$ and the restriction of a map to any subdomain. The class Möb of Möbius mappings acting in ${{\mathbf {R}}^n}$ is of particular interest. Assume that the class $F$ is "$c$-uniformly close" to Möb. Then we show that any map in $F$ is either constant or a local quasiconformal homeomorphism. As a corollary we obtain a distinctly elementary proof of the Local Injectivity Theorem for quasiregular mappings.