We classify the harmonic morphisms with one-dimensional fibres (1) from real-analytic conformally-flat Riemannian manifolds of dimension at least four, and (2) between conformally-flat Riemannian manifolds of dimensions at least three.

Let (M, g) be a smooth compact Riemannian manifold without boundary of dimension n ≥ 6. We prove that ‖u‖ L2 ∗ (M,g) ≤ K 2 ∫ M { |∇gu| 2 + c(n)Rgu 2 } dvg + A‖u‖ 2 L2n/(n+2)(M,g), for all u ∈ H(M), where 2 = 2n/(n − 2), c(n) = (n − 2)/[4(n − 1)], Rg is the scalar curvature, K −1 = inf ‖∇u‖L2(Rn)‖u‖ −1 L2n/(n−2)(Rn) and A > 0 is a constant depending on (M, g) only. The inequality is sharp in the...