JournalscmhVol. 77, No. 1pp. 39–77

Harmonic forms and near-minimal singular foliations

  • G. Katz

    Framingham, USA
Harmonic forms and near-minimal singular foliations cover
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Abstract

For a closed 1-form ω\omega with Morse singularities, Calabi discovered a simple global criterion for the existence of a Riemannian metric in which ω\omega is harmonic. For a codimension 1 foliation F\cal {F} , Sullivan gave a condition for the existence of a Riemannian metric in which all the leaves of F\cal {F} are minimal hypersurfaces. The conditions of Calabi and Sullivan are strikingly similar. If a closed form ω\omega has no singularities, then both criteria are satisfied and, for an appropriate choice of metric, ω\omega is harmonic and the associated foliation Fω\cal {F}_\omega is comprised of minimal leaves. However, when ω\omega has singularities, the foliation Fω\cal {F}_\omega is not necessarily minimal.¶ We show that the Calabi condition enables one to find a metric in which ω\omega is harmonic and the leaves of the foliation are minimal outside a neighborhood U of the ω\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 Fω\cal {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.

Cite this article

G. Katz, Harmonic forms and near-minimal singular foliations. Comment. Math. Helv. 77 (2002), no. 1, pp. 39–77

DOI 10.1007/S00014-002-8331-5