Yamabe metrics of positive scalar curvature and conformally flat manifolds (Q1339909)

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.