In this paper we present a necessary and sufficient condition for a C^2 Teichmüller mapping to be $\rho$-harmonic. By this result we show that there is no solution to the Schoen conjecture in the class of C^2 Teichmüller mappings. We also obtain two characterizations for $\pi$-harmonic Teichmüller mappings.
Xingdi Chen. Ainong Fang. "Harmonic Teichmüller mappings." Proc. Japan Acad. Ser. A Math. Sci. 82 (7) 101 - 105, March 2006. https://doi.org/10.3792/pjaa.82.101