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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.

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