Quantitative gradient estimates for harmonic maps into singular spaces

Abstract

In this paper, we will show the Yau's gradient estimate for harmonic maps into a metric space $(X,d_X)$ with curvature bounded above by a constant $\kappa$, $\kappa \ge 0$, in the sense of Alexandrov. As a direct application, it gives some Liouville theorems for such harmonic maps. This extends the works of S. Y. Cheng [4] and H. I. Choi [5] to harmonic maps into singular spaces.

The main results in the first version have been improved. To appear in SCIENCE CHINA Mathematics

References in corpus (2)

• Liouville theorems for $f$-harmonic maps into Hadamard spaces
• A Liouville-type theorem and Bochner formula for harmonic maps into metric spaces