Transversality theorems for harmonic forms
type: Item
instance of: scholarly article
title: Transversality theorems for harmonic forms
zbMATH Open document ID: 1071.58002
author: Ko Honda
published in: Rocky Mountain Journal of Mathematics
publication date: 2004-10-01T00:00:00Z
review text: 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.
reviewed by: Jesús A. Álvarez López
Mathematics Subject Classification ID: 58A14
Mathematics Subject Classification ID: 58J50
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 58J37
zbMATH DE Number: 2104655
zbMATH Keywords: harmonic forms
zbMATH Keywords: transversality
zbMATH Keywords: zero set
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181069872
OpenAlex ID: W1977437956
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DOI: 10.1216/RMJM/1181069872
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description: scientific article; zbMATH DE number 2104655