Journal of the
Korean Mathematical Society JKMS

Let $M$ be a complete Riemannian manifold and let $N$ be a Riemannian manifold of nonpositive scalar curvature. Let $\mu_0$ be the least eigenvalue of the Laplacian acting on $L^2$-functions on $M$. We show that if $Ric^M \geq -\mu_0$ at all $x\in M$ and either $Ric^M>-\mu_0$ at some point $x_0$ or ${\rm Vol}(M)$ is infinite, then every harmonic morphism $\phi:M\to N$ of finite energy is constant.

Keywords: harmonic map, harmonic morphism

MSC numbers: 5.80E+21