A gluing theorem for quasiconformal mappings

Abstract

We prove, by using the main inequality of Reich and Strebel, that any n K-quasiconformal germs defined on n disjoint domains in the Riemann sphere can be glued by one (K + ε)-quasiconformal homeomorphism, where ε is a positive number which can go to zero as the domains of germs shrinking to n points. This generalizes a result in [8] where only the case K = 1 has been considered.