Stable harmonic maps into the complex projective spaces (Q1915928)

A main result obtained in this paper is that if \((M,g)\) is a compact \(n\)-dimensional Riemannian manifold without boundary and \(f\) is a smooth stable harmonic map from \(M\) into \(\mathbb{C} P^k\) with the Fubini-Study metric, then the rank of the differential of \(f\) is even at any point in \(M\). Additionally, any smooth stable harmonic map into \(S^2\) is a harmonic morphism. In the case of compact 3-manifolds with \(f\) a non-constant stable harmonic map into \(\mathbb{C} P^k\) (with the Fubini-Study metric), the image set \(f(M)\) is a compact algebraic curve whose singular locus consists of finitely many points. In the last section of the paper the author shows that there exists a topological obstruction for the existence of a stable harmonic map from \(M^3\) into \(\mathbb{C} P^k\). That is, \(M^3\) is necessarily a Seifert fibre space.