Harmonic forms and near-minimal singular foliations

G. Katz
2002 Commentarii Mathematici Helvetici  
For a closed 1-form ω with Morse singularities, Calabi discovered a simple global criterion for the existence of a Riemannian metric in which ω is harmonic. For a codimension 1 foliation F , Sullivan gave a condition for the existence of a Riemannian metric in which all the leaves of F are minimal hypersurfaces. The conditions of Calabi and Sullivan are strikingly similar. If a closed form ω has no singularities, then both criteria are satisfied and, for an appropriate choice of metric, ω is
more » ... monic and the associated foliation F ω is comprised of minimal leaves. However, when ω has singularities, the foliation Fω is not necessarily minimal. We show that the Calabi condition enables one to find a metric in which ω is harmonic and the leaves of the foliation are minimal outside a neighborhood U of the ω-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 F ω outside U remains stable. Furthermore, all compact leaves missing U are volume minimizing cycles in their homology classes. Their volumes are controlled explicitly. Remark. Locally, ω = df , f being a Morse function. Property (i) prevents f from having local maxima and minima in the interior of M and (ii) -on its boundary. A closed Morse-type 1-form which possesses properties (i) and (ii) will be called a Calabi form. Vol. 77 (2002) Harmonic forms and near-minimal singular foliations 41 42 G. Katz CMH In small dimensions, these foliations definitely fail to be minimal with respect to any metric. Nevertheless, we suspect that, if the dimension of M exceeds 7, sometimes the word "near" can be be dropped from the statement about the leaves of F ω (cf. Conjecture 4.1). Many results of this paper on the volume-minimizing cycles (built of compact F ω -leaves) have somewhat more general classical analogs, formulated in terms the mass-minimizing foliation cycles and currents. These classical results were established for generic non-singular foliations on compact closed manifolds (cf. [HL], [S], [S1]) and stated in terms of the Geometric Measure Theory. In this context, our contribution can be described as reproducing these results for the non-compact manifolds {M ω } in a fashion that permits an extension of the metric in question across the singularities. In particular, we prove (cf. Proposition 2.5) a "taut" version of the rational Poincaré duality (in dimensions 1 and (n − 1)). We also extend the setting for manifolds with boundary and investigate the impact of boundary effects on the intrinsic harmonicity and minimality of the leaves. When the form ω| ∂M is non-singular, and ω satisfies (0.2), then it is possible to "synchronize" the harmonicity and minimality on M and ∂M ; for ω| ∂M with singularities, it is very much an open problem. The introduction of foliations with singularities, induced by closed 1-forms, drastically changes the landscape of the classical foliation theory. On the one hand, one avoids some of the pathologies (think about the Reeb foliation), characteristic for the most general (non-singular) foliations, on the other hand, the presence of singularities generates new diverse possibilities and complications. For example, Novikov's Theorem ([N2]) states that, if a foliation F ω is produced by a closed non-singular form ω, then all the leaves are either compact, or non-compact. This is not the case for foliations generated by closed forms with singularities -such foliations often are mixed bags. The objects and constructions that facilitate our proofs are, so to speak, handmade. As a result, we are able to avoid a great deal of Functional Analysis and Geometric Measure Theory. Our only generic tool is Stokes' Theorem.
doi:10.1007/s00014-002-8331-5 fatcat:rmu4qyl7grcmrcrdnjwuhnnwmu