Lipschitz regularity of energy-minimal mappings between doubly connected Riemann surfaces

Let $M$ and $N$ be doubly connected Riemann surfaces with $\mathscr{C}^{1,\alpha}$ boundaries and with nonvanishing conformal metrics $\sigma$ and $\wp$ respectively, and assume that $\wp$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. Assume that ${\Ho}_\rho(M, N)$ is the class of all $\mathscr{W}^{1,2}$ bomeomorphisms between $M$ and $N$ and assume that $\mathcal{E}^\wp: \overline{\Ho}_\rho(M, N)\to \mathbf{R}$ is the Dirichlet-energy functional, where $\overline{\Ho}_\rho(M, N)$ is the closure of ${\Ho}_\rho(M, N)$ in $\mathscr{W}^{1,2}(M,N)$. By using a result of Iwaniec, Kovalev and Onninen in \cite{duke} that the minimizer, is locally Lipschitz, we prove that the minimizer, of the energy functional $\mathcal{E}^\wp$, which is not a diffeomorphism in general, is a globally Lipschitz mapping of $M$ onto $N$.