Quasiconformal mappings u:Ω->Ω between open domains in Rn, are W{1,n} homeomorphisms whose dilation K=|du|/ (det du)1/n is in L∞. A classical problem in geometric function theory consists in finding QC minimizers for the dilation within a given homotopy class or with prescribed boundary data. In a joint work with A. Raich we study C2 extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a `localized' form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.