Harmonic forms and near-minimal singular foliations
A closed 1-form \(\omega\) (\(d\omega = 0\)) on a smooth manifold \(M\) is called \textit{Morse} if \(M\) can be covered by open sets \(\mathcal U =\{U\}\) so that the restriction of \(\omega\) on each \(U\) is the differential of a Morse function \(f_U: U\to \mathbb R\), i.e. \(\omega|_U = df_U\). A closed 1-form \(\omega\) is \textit{intrinsically harmonic}, if there exists a Riemannian metric \(g\) on \(M\) such that \(\omega\) is harmonic with respect to \(g\). \textit{E. Calabi} in his paper ``An intrinsic characterization of harmonic 1-forms''\ [in: ``Global Analysis, Papers in Honor of K. Kodaira'', 101-117 (1969; Zbl 0194.24701)] has found geometric conditions which are equivalent to the intrinsic harmonicity. A smooth path \(\gamma: [0,1]\to M\) is called \(\omega\)-positive if \(\omega(\dot \gamma(t)) > 0\) for all \(t\in [0,1]\). The theorem of Calabi states: A closed 1-form \(\omega\) of the Morse type is intrinsically harmonic if and only if (1) \(\omega\) has no zeros with the Morse indices \(0\) and \(n = \dim M\); and (2) through any non-singular point \(p\in M\) there exists a closed \(\omega\)-positive path. The paper under review strengthens Calabi's theorem in several directions. In a simplified form the result can be formulated as follows: Theorem: Let \(M\) be a closed smooth manifold and let \(\omega\) be a closed, intrinsically harmonic 1-form on \(M\) with the Morse-type singularities. Let \(\mathcal F_\omega\) denote the singular \((n-1)\)-dimensional foliation generated by the kernels of \(\omega\). Then, for any neighborhood \(V\subset M\) of the zero set of \(\omega\), there exists a Riemannian metric \(g\) on \(M\) such that: (1) the form \(\omega\) is harmonic with respect to \(g\), and (2) all leaves of the foliation \(\mathcal F_\omega\) are minimal hypersurfaces in \(M-V\) with respect to \(g\). This result generalizes a theorem of \textit{D. Sullivan} [Comment. Math. Helv. 54, 218-223 (1979; Zbl 0409.57025)] who studied foliations without singularities. The author also describes conditions under which the compact leaves of \(\mathcal F_\omega\) are volume-minimizing cycles in their homology classes. The actual results proven in the paper are much more general and informative; in particular, they allow manifolds with boundary. The paper contains a discussion of open problems and some interesting conjectures.
scientific article; zbMATH DE number 1756818