Energy-minimal diffeomorphisms between doubly connected Riemann surfaces

Let (M) and (N) be doubly connected Riemann surfaces with boundaries and with nonvanishing conformal metrics (sigma ) and (rho ) respectively, 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 (M) and (N) that minimize the Dirichlet energy. Among all homeomorphisms (f :M{overset{{}_{ tiny {mathrm{onto}} }}{longrightarrow }} N) between doubly connected Riemann surfaces such that ({{mathrm{Mod,}}}M leqslant {{mathrm{Mod,}}}N) there exists, unique up to conformal automorphisms of M, an energy-minimal diffeomorphism which is a harmonic diffeomorphism. The results improve and extend some recent results of Iwaniec et al. (Invent Math 186(3):667–707, 2011), where the authors considered bounded doubly connected domains in the complex plane w.r. to Euclidean metric.