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Yamabe metrics of positive scalar curvature and conformally flat manifolds. (English) Zbl 0810.53030
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.

53C20 Global Riemannian geometry, including pinching
58D17 Manifolds of metrics (especially Riemannian)
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