Quantitative gradient estimates for harmonic maps into singular spaces

In this paper, we show the Yau’s gradient estimate for harmonic maps into a metric space ( X , d X ) with curvature bounded above by a constant κ ( κ ⩾ 0) in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of Cheng (1980) and Choi (1982) to harmonic maps into singular spaces.