The local homeomorphism property of spatial quasiregular mappings with distortion close to one

We give a quantitative proof for a theorem of Martio, Rickman and V?is?l? [MRV] on the rigidity of the local homeomorphism property of spatial quasiregular mappings with distortion close to one. The proof is based on a distortion theory established by using two main tools. First, we use a conformal invariant between sphere families and components of their preimages under quasiregular mappings. Secondly, we use Hall’s quantitative isoperimetric inequality result [H] to relate two different types of distortion. Received: April 2004 Revision: October 2004 Accepted: December 2004