Quantitative gradient estimates for harmonic maps into singular spaces

TL;DR

This paper extends Yau's gradient estimate to harmonic maps into singular metric spaces with curvature bounds, leading to new Liouville theorems and broadening classical results to more general geometric contexts.

Contribution

It generalizes Yau's gradient estimate to harmonic maps into Alexandrov spaces with curvature bounds, providing new Liouville theorems for such maps.

Findings

Yau's gradient estimate is valid for harmonic maps into singular spaces with curvature bounded above.

The results lead to Liouville theorems for harmonic maps into these spaces.

Extension of classical harmonic map estimates to Alexandrov 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 $\kappa$, $\kappa\geq0$, 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.