Mapping of Least ρ-Dirichlet Energy between Doubly Connected Riemann Surfaces

Abstract

In this note, we consider the mappings $h:h:\mathbb{X}\rightarrow\mathbb{Y}$ between doubly connected Riemann surfaces having least ρ-Dirichlet energy. For a pair of doubly connected Riemann surfaces, in which $\mathbb{X}$ has finite conformai modulus, we establish the following principle: A mapping h in the class${\overline H _2}\left(\mathbb{X}{,}\mathbb{Y}\right)$of strong limits of homeomorphisms in Sobolev space${\mathbb{W}^{1,2}}\left(\mathbb{X}{,}\mathbb{Y}\right)$is ρ-energy-minimal if and only if its Hopf-differential is analytic in$\mathbb{X}$and real along$\partial\mathbb{X}$. It improves and extends the result of Iwaniec et al. (see Theorem 1.4 in [Arch. Ration. Mech. Anal., 209, 401–453 (2013)]). Furthermore, we give an application of the principle. Any ρ-energy minimal diffeomorphism is ρ-harmonic, however, we give a $1/{\left| w \right|^2}$-harmonic diffemorphism which is not $1/{\left| w \right|^2}$-energy minimal diffeomorphism. At last, we investigate the necessary and sufficient conditions for the existence of $1/{\left| w \right|^2}$-harmonic mapping from doubly connected domain Ω to the circular annulus A(1, R).