Distortion in the spherical metric under quasiconformal mappings

Abstract:

This paper contains bounds for the distortion in the spherical metric, that is to say, bounds for the constant of Hölder continuity of mappings $f \colon ({\mathbb R}^n,q) \to ({\mathbb R}^n, q)$ where $q$ denotes the spherical metric. The mappings considered are $K$-quasiconformal ($K\ge 1$) and satisfy some normalizations or restrictions. All bounds are explicit and asymptotically sharp as $K \to 1$.