For a closed 1-form with Morse singularities, Calabi discovered a simple global criterion for the existence of a Riemannian metric in which is harmonic. For a codimension 1 foliation , Sullivan gave a condition for the existence of a Riemannian metric in which all the leaves of are minimal hypersurfaces. The conditions of Calabi and Sullivan are strikingly similar. If a closed form has no singularities, then both criteria are satisfied and, for an appropriate choice of metric, is harmonic and the associated foliation is comprised of minimal leaves. However, when has singularities, the foliation is not necessarily minimal.¶ We show that the Calabi condition enables one to find a metric in which is harmonic and the leaves of the foliation are minimal outside a neighborhood U of the -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 outside U remains stable. Furthermore, all compact leaves missing U are volume minimizing cycles in their homology classes. Their volumes are controlled explicitly.
Cite this article
G. Katz, Harmonic forms and near-minimal singular foliations. Comment. Math. Helv. 77 (2002), no. 1, pp. 39–77