  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}.
