Collapsing and monopole classes of 3-manifolds
type: Item
instance of: scholarly article
title: Collapsing and monopole classes of 3-manifolds
zbMATH Open document ID: 1117.57031
author: Chanyoung Sung
published in: Journal of Geometry and Physics
publication date: 2007-02-13T00:00:00Z
review text: 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.
reviewed by: David Auckly
Mathematics Subject Classification ID: 57R57
Mathematics Subject Classification ID: 57M50
Mathematics Subject Classification ID: 53C99
zbMATH DE Number: 5125067
zbMATH Keywords: Seiberg-Witten
zbMATH Keywords: monopole class
zbMATH Keywords: Yamabe invariant
MaRDI profile type: Publication
full work available at URL: j.geomphys.2006.04.006
OpenAlex ID: W2056873115
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DOI: 10.1016/J.GEOMPHYS.2006.04.006
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description: scientific article; zbMATH DE number 5125067