An Aronsson type approach to extremal quasiconformal mappings

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.