Lipschitz property of minimisers between double connected surfaces

We study the global Lipschitz character of minimisers of the Dirichlet energy of diffeomorphisms between doubly connected domains with smooth boundaries from Riemann surfaces. The key point of the proof is the fact that minimisers are certain Noether harmonic maps, with Hopf differential of special form, a fact invented by Iwaniec, Koh, Kovalev and Onninen in \cite{inv} for Euclidean metric and by the author in \cite{calculus} for the arbitrary metric, which depends deeply on a result of Jost \cite{Job1}.