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

In this note,we consider the mappings h:X→Y between doubly connected Riemann surfaces having leastρ-Dirichlet energy.For a pair of doubly connected Riemann surfaces,in which X has finite conformal modulus,we establish the following principle:A mapping h in the class H2(X,Y)of strong limits of homeomorphisms in Sobolev space W1,2(X,Y)isρ-energy-minimal if and only if its Hopf-differential is analytic in X and real along?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.Anyρ-energy minimal diffeomorphism isρ-harmonic,however,we give a 1/|w|~2-harmonic diffemorphism which is not 1/|w|~2-energy minimal diffeomorphism.At last,we investigate the necessary and sufficient conditions for the existence of 1/|w|~2-harmonic mapping from doubly connected domainΩto the circular annulus A(1,R).