Collapsing and monopole classes of 3-manifolds
Seiberg-Witten theory has produced many interesting results about smooth \(4\)-manifolds and a few about topological \(3\)-manifolds. It can also produce interesting results about the Riemannian geometry of \(3\)-manifolds. This paper presents a couple of results in this direction. Different geometric applications appear in \textit{M. Itoh} and \textit{T. Yamase} [Osaka J. Math 43, 121--129 (2006; Zbl 1107.57021)]. The first result is a translation of LeBrun's curvature estimates from the \(4\)-dimensional case to the \(3\)-dimensional case. The second theorem has a slightly technical statement and a more substantial proof, but concludes that there is a sequence of metrics on any fibered \(3\)-manifold such that the product of corresponding \(L^2\) norms of the Ricci curvature and a \(1\)-form pulled back from the base converge to a specific function of the cohomology class of the form. There is a similar result for the scalar curvature. An application of this theorem restricts the possible monopole classes on a \(3\)-manifold with a fibered summand. The author also derives a constraint on the Yamabe invariant from monopole classes. This in turn shows that any sequence of metrics for which the \(L^2\) norm of the positive part of the scalar curvature tends to zero has injectivity radius converging to zero.
scientific article; zbMATH DE number 5125067
scientific article