Energy-minimal diffeomorphisms between doubly connected Riemann surfaces

Abstract

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.