Yamabe metrics of positive scalar curvature and conformally flat manifolds
type: Item
instance of: scholarly article
title: Yamabe metrics of positive scalar curvature and conformally flat manifolds
zbMATH Open document ID: 0810.53030
DOI: 10.1016/0926-2245(94)00015-8
published in: Differential Geometry and its Applications
publication date: 1995-04-09T00:00:00Z
review text: Let \({\mathcal C}{\mathcal Y} (n, \mu_ 0, R_ 0)\) be the class of compact connected smooth manifolds \(M\) of dimension \(n \geq 3\) and with Yamabe metrics \(g\) of unit volume such that each \((M,g)\) is conformally flat and satisfies \[ \mu (M, [g]) \geq \mu_ 0 > 0,\quad \int_ M | \widehat{\text{Ric}}_ g| ^{n/2} dv_ g \geq R_ 0, \] where \([g]\), \(\mu(M,[g])\) and \(\widehat {\text{Ric}}_ g\) denote the conformal class of \(g\), the Yamabe invariant of \((M, [g])\) and the traceless part of the Ricci tensor of \(g\), respectively. In this paper, we study the boundary \(\partial {\mathcal C} {\mathcal Y} (n, \mu_ 0, R_ 0)\) of \({\mathcal C} {\mathcal Y} (n, \mu_ 0, R_ 0)\) in the space of all compact metric spaces equipped with the Hausdorff distance. We prove that an element in \(\partial {\mathcal C} {\mathcal Y} (n, \mu_ 0, R_ 0)\) is a compact metric space \((X, d)\). In particular, if \((X, d)\) is not a point, then it has a structure of smooth manifold outside a finite subset \(\mathcal S\), and moreover, on \(X \setminus {\mathcal S}\) there is a conformally flat metric \(g\) of positive constant scalar curvature which is compatible with the distance \(d\). As an application, we also give a pinching theorem for spherical space forms. The crucial point is to prove new geometric inequalities for Yamabe metrics of positive scalar curvature.
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 58D17
zbMATH DE Number: 701733
zbMATH Keywords: Yamabe metrics
zbMATH Keywords: conformally flat
zbMATH Keywords: Yamabe invariant
zbMATH Keywords: geometric inequalities
zbMATH Keywords: positive scalar curvature
Wikidata QID: Q115362629
author: Kazuo Akutagawa
MaRDI profile type: Publication
full work available at URL: 0926-2245(94)00015-8
OpenAlex ID: W2085161880
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description: scientific article; zbMATH DE number 701733