Liouville theorems for harmonic maps into CAT(1) spaces

We prove Liouville theorems for harmonic maps into metric spaces with curvature bounded above by a constant k>0 in the sense of Alexandrov. Our results generalize a Liouville theorem for harmonic maps from manifolds with non-negative Ricci curvature into CAT(1) space in the recent work (Zhang et al. in Sci China Math 62(11): 2371-2400, 2019) of Zhang-Zhong-Zhu. As a direct corollary, we improve a well-known result of Choi (Proc Amer Math Soc 85(1): 91-94, 1982) for harmonic maps between smooth Riemannian manifolds.