An Aronsson type approach to extremal quasiconformal mappings

TL;DR

This paper investigates extremal quasiconformal mappings in three-dimensional space, establishing conditions for localized extremality and demonstrating short-term existence of smooth solutions to related gradient flows.

Contribution

It introduces a new Aronsson-type framework for extremal quasiconformal mappings and proves short time existence results for associated gradient flows.

Findings

Necessary and sufficient conditions for localized extremality.

Existence of smooth solutions to the gradient flow.

Extension of Aronsson's ideas to quasiconformal mappings.

Abstract

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.