Equivalent characterizations of harmonic Teichmüller mappings

In this paper, three equivalent conditions of $\rho$-harmonic Teichmüller mapping are given firstly. As an application, we investigate the relationship between a $\rho$-harmonic Teichmüller 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 $1/|\omega {|}^2$-harmonic Teichmüller mappings is given as a by-product. Our results extend the corresponding researches of harmonic Teichmüller mappings.