A Liouville theorem for harmonic maps (Q2732551)

By using a method of \textit{P. Li} and \textit{J. P. Wang} [J. Diff. Geom. 48, 497-530 (1998; Zbl 0926.58011)], the author obtains the following result: Let \(M,N\) be complete Riemannian manifolds. \(M\) admits no nonconstant bounded harmonic functions, and the sectional curvature of \(N\) has upper bound \(k>0\). Let \(u:M\to N\) be a harmonic map with \(u(M)\subset B_R(P) \), where \(R=\pi/2 \sqrt k\), \(P\in N\). If \(B_R(p)\) lies in the cut-locus of \(P\), and \(\overline {u(M)} \cap\partial B_R(p)\) contains at most one point, then \(u\) must be constant.NEWLINENEWLINENEWLINEThis generalizes a result of \textit{W. Kendall} [Proc. Lond. Math. Soc., III. Ser. 61, 371-406 (1990; Zbl 0675.58042)] which instead assumes that \(R<\pi/2 \sqrt k\).