Commentarii Mathematici Helvetici

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Volume 77, Issue 1, 2002, pp. 39–77
DOI: 10.1007/s00014-002-8331-5

Published online: 2002-03-31

Harmonic forms and near-minimal singular foliations

G. Katz[1]

(1) Framingham, USA

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 $ \cal {F} $, Sullivan gave a condition for the existence of a Riemannian metric in which all the leaves of $ \cal {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 $ \cal {F}_\omega $ is comprised of minimal leaves. However, when $ \omega $ has singularities, the foliation $ \cal {F}_\omega $ is not necessarily minimal.ΒΆ 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 $ \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.

Keywords: Closed 1-forms, intrinsic harmonicity, minimal foliations, volume-minimizing cycles, Morse-type singularities

Katz G.: Harmonic forms and near-minimal singular foliations. Comment. Math. Helv. 77 (2002), 39-77. doi: 10.1007/s00014-002-8331-5