Energy-minimal diffeomorphisms between doubly connected Riemann surfaces

TL;DR

This paper proves the existence and uniqueness of energy-minimizing harmonic diffeomorphisms between doubly connected Riemann surfaces with smooth metrics, extending previous results in the complex plane to more general settings.

Contribution

It establishes the existence and uniqueness of energy-minimal harmonic diffeomorphisms between doubly connected Riemann surfaces with bounded curvature metrics, generalizing prior Euclidean results.

Findings

Existence of energy-minimal harmonic diffeomorphisms.

Uniqueness up to conformal automorphisms.

Extension of previous Euclidean domain results to curved surfaces.

Abstract

Let $N=(\Omega,\sigma)$ and $M=(\Omega^*,\rho)$ 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 $\Omega$ and $\Omega^*$ that minimize the Dirichlet energy. In the class of all homeomorphisms $f \colon \Omega \onto \Omega^\ast$ between doubly connected domains such that $\Mod \Omega \le \Mod \Omega^\ast$ there exists, unique up to conformal authomorphisms of $\Omega$, 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.