Transversality theorems for harmonic forms
For a closed oriented manifold, the author studies the regularity and transversality properties of the zero set of harmonic forms for generic Riemannian metrics. For instance, he shows the following results by using transversality theory. Let \(M\) be a closed, oriented \(4\)-manifold with \(b^{+}_2(M)>0\). Let \(\text{ Met}^l(M)\) be the space of \(C^l\)-Hölder metrics on \(M\) for a large enough non-integer \(l\). Let \(Q^+\subset H^2(M;\mathbb{R})\times\text{ Met}^l(M)\) be the Banach submanifold consisting of pairs \(([\omega],g)\) such that \(\omega\) is a self-dual \(g\)-harmonic \(2\)-form with \([\omega]\neq 0\). Then there is a dense open set \({\mathcal U}\subset Q^+\) such that, if \(([\omega],g)\in {\mathcal U}\) with \(\omega\) \(g\)-harmonic, then \(\omega\) has regular zero set (it consists of disjoint circles). Let \(M\) be a closed, oriented \(4\)-manifold with \(b^{\pm}_2(M)>0\), and let \(\text{ Met}^l(M)\) be as above. Then there exists a dense open set \({\mathcal U}\subset H^2(M;\mathbb{R})\times\text{ Met}^l(M)\) such that if \(([\omega],g)\in {\mathcal U}\) with \(\omega\) \(g\)-harmonic, then \(\omega\) is neither self-dual nor anti-self-dual, and \(\omega\) has no zeros, has full rank away from a submanifold of codimension \(1\), and is self-dual/anti-self-dual on a union of disjoint circles. The author also considers the Dirichlet problem on a bounded domain in \({\mathbb R}^n\) and proves that, for a fixed boundary condition and a generic choice of metric, the solution has regular zero set.
scientific article; zbMATH DE number 2104655
scientific article