Equivalent characterizations of harmonic Teichmuller mappings

In this paper, three equivalent conditions of rho-harmonic Teichmuller mapping are given firstly. As an application, we investigate the relationship between a rho-harmonic Teichmuller mapping and its associated holomorphic quadratic differential and obtain a relatively simple method to prove Theorem 2.1 in [1] Furthermore, the representation theorem of 1/vertical bar omega vertical bar(2)-harmonic Teichmuller mappings is given as a by-product. Our results extend the corresponding researches of harmonic Teichmuller mappings.