A note on the rho-Nitsche conjecture

摘要

Let be a radial symmetric Riemannian metric defined in the annulus A(1, R). Kalaj conjectured that if there exists a -harmonic homeomorphism from the annulus A(1, r) onto A(1, R), then R satisfies the following -Nitsche condition In this note, we prove that if , then there exists a -harmonic mapping between A(1, r) and A(1, R) if and only if , i.e. the -Nitsche condition holds. This gives a positive answer to Kalaj's conjecture for . Some related topics are also discussed.