Yamabe metrics of positive scalar curvature and conformally flat manifolds

AbstractLet CY(n,μ, R0 be the class of compact connected smooth manifolds M of dimension n ⩾ 3 and with Yamabe metrics g of unit volume such that each (M, g) is conformally flat and satisfies μ(M,[g]) ⩾μ0 > 0, ∫M|Eg|n2dvg⩽R0, where [g], μ(M,[g]) and Eg 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 ACY(n, μ0, R0 of CY(n, μ0, R0) in the space of all compact metric spaces equipped with the Hausdorff distance. We shall show that an element in ACY(n, μ0, R0) 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 S, and moreover, on F\S there is a conformally flat metric g of positive constant scalar curvature which is compatible with the distance d