Bernstein theorem for harmonic morphisms from \((N+1)\)-sphere to \(N\)-manifold (Q1365356)

Harmonic morphisms may be characterized as harmonic maps which are horizontally weakly conformal. In this paper, the author proves the following Bernstein theorem for harmonic morphisms. If \(\varphi: (S^{n+1}, g_{can}) \to N^n\) \((n\geq 3)\) is a nonconstant harmonic morphism, then \(\varphi\) is, alternating an orthogonal change of coordinates, the Hopf fibration \(\pi: S^{n+1} \to \mathbb{C} P^{n/2}\) followed by a homothetic map. In particular, this implies that \(N^n= \mathbb{C} P^{n/2}\). This may be viewed as a generalization of the theorem by \textit{P. Baird} and \textit{J. C. Wood} [Math. Ann. 280, No. 4, 579-603 (1988; Zbl 0621.58011)].