Four-manifolds with harmonic 2-forms of constant length

Abstract

It was shown by Seaman that if a compact, connected, oriented, riemannian 4-manifold (M, g) of positive sectional curvature admits a harmonic 2-form of constant length, then M has definite intersection form and such a harmonic form is unique up to constant multiples. In this paper, we show that such a manifold is diffeomorphic to $$\mathbb {CP}_{2}$$ with a slightly weaker curvature hypothesis and there is an infinite dimensional moduli space of such metrics near the Fubini-Study metric on $$\mathbb {CP}_{2}$$ .