Abstract
Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that Ric M ⩾ − 4 ( p − 1 ) p 2 μ 0 at all x ∈ M and > − 4 ( p − 1 ) p 2 μ 0 at some point x 0 ∈ M , where μ 0 > 0 is the least eigenvalue of the Laplacian acting on L 2 -functions on M. Let 2 ⩽ q ⩽ p . Then any q-harmonic map ϕ : M → N of finite q-energy is constant. Moreover, if N is a Riemannian manifold of non-positive scalar curvature, then any q-harmonic morphism ϕ : M → N of finite q-energy is constant.
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