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Quasiconformal mappings $u:\Omega\to \Omega'$Ìýbetween open domains in $R^n$,Ìýare $ W^{1,n}$ homeomorphisms whose dilation $K=|du|/ (det du)^1/n$ is in $L^\infty$. A classical problem in geometric function theory consists in finding QC minimizers for the dilation withinÌýa given homotopy class or with prescribed boundary data.ÌýIn a joint work with A. Raich we study $C^2$ 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.