Energy-minimal diffeomorphisms between doubly connected Riemann surfaces

Let $N=(\mathrm{\Omega },\sigma )$ and $M=({\mathrm{\Omega }}^{\ast },\rho )$ be doubly connected Riemann surfaces and assume that $\rho$ is a smooth metric with bounded Gauss curvature $\mathcal{K}$ and finite area. The paper establishes the existence of homeomorphisms between $\mathrm{\Omega }$ and ${\mathrm{\Omega }}^{\ast }$ that minimize the Dirichlet energy. In the class of all homeomorphisms $f\text{\hspace{0.17em}⁣}:\mathrm{\Omega }\text{onto}{\mathrm{\Omega }}^{\ast }$ between doubly connected domains such that $\text{Mod}\mathrm{\Omega }\le \text{Mod}{\mathrm{\Omega }}^{\ast }$ there exists, unique up to conformal authomorphisms of $\mathrm{\Omega }$, an energy-minimal diffeomorphism which is a harmonic diffeomorphism. The results improve and extend some recent results of Iwaniec, Koh, Kovalev and Onninen (Inven. Math. (2011)), where the authors considered doubly connected domains in the complex plane w.r. to Euclidean metric.