Some rigidity theorems of harmonic maps between Finsler manifolds (Q2874718)

The authors study the properties of harmonic maps between Finsler manifolds. Some rigidity theorems for such maps are proved. The main results of the paper are: NEWLINENEWLINENEWLINE(1) If \(\phi\) is a conformal harmonic map from an \(n(>2)\)-dimensional Finsler manifold \((M,F)\) to a Finsler manifold \((\widetilde{M},\widetilde{F})\), then \(\phi\) must be homothetic. NEWLINENEWLINENEWLINE(2) Any nondegenerate harmonic map from a compact Riemannian manifold without boundary with non-negative Ricci curvature to a Berwald manifold with non-positive flag curvature must be totally geodesic. NEWLINENEWLINENEWLINE(3) Let \((M,g)\) be an \(n(>2)\)-dimensional compact Riemannian manifold without boundary whose Ricci curvature is \(\text{Ric}^M\geq \lambda\) and \((\widetilde{M},\widetilde{F})\) a Berwald manifold whose flag curvature is \(K^{\widetilde{M}}\leq\mu\), where \(\lambda, \mu\) are two positive constants. For any nondegenerate smooth map \(\phi:(M,g)\longrightarrow (\widetilde{M},\widetilde{F})\), if its energy density satisfies \(e(\phi)-\frac{1}{n}\,\overline{e}(\phi)\leq \frac{\lambda}{2\mu}\), where \(\overline{e}(\phi)\) is the mean energy density of \(\phi\), then \(\phi\) must be a totally geodesic conformal map and \(e(\phi)\) is equal to the constant \(\frac{n\lambda}{2(n-1)\mu}\).