Harmonic forms and near-minimal singular foliations

Abstract

For a closed 1-form $\omega$ with Morse singularities, Calabi discovered a simple global criterion for the existence of a Riemannian metric in which $\omega$ is harmonic. For a codimension 1 foliation $\mathcal{F}$ , Sullivan gave a condition for the existence of a Riemannian metric in which all the leaves of $\mathcal{F}$ are minimal hypersurfaces. The conditions of Calabi and Sullivan are strikingly similar. If a closed form $\omega$ has no singularities, then both criteria are satisfied and, for an appropriate choice of metric, $\omega$ is harmonic and the associated foliation ${\mathcal{F}}_{\omega }$ is comprised of minimal leaves. However, when $\omega$ has singularities, the foliation ${\mathcal{F}}_{\omega }$ is not necessarily minimal.¶ We show that the Calabi condition enables one to find a metric in which $\omega$ is harmonic and the leaves of the foliation are minimal outside a neighborhood U of the $\omega$ -singular set. In fact, we prove the best possible result of this type: we construct families of metrics in which, as U shrinks to the singular set, the taut geometry of the foliation ${\mathcal{F}}_{\omega }$ outside U remains stable. Furthermore, all compact leaves missing U are volume minimizing cycles in their homology classes. Their volumes are controlled explicitly.