YAMABE METRICS OF POSITIVE SCALAR CURVATURE AND CONFORMALLY FLAT MANIFOLDS

Let CY(n, mu0, R0) be the class of compact connected smooth manifolds M of dimension n greater-than-or-equal-to 3 and with Yamabe metrics g of unit volume such that each (M, g) is conformally flat and satisfies mu(M,[g]) greater-than-or-equal-to mu0 > 0, integral(M)E(g)\n/2dupsilon(g) less-than-or-equal-to R0, where [g], mu(M,[g]) and E(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-derivativeCY(n,mu0, R0) of CY(n, mu0, R0) in the space of all compact metric spaces equipped with the Hausdorff distance. We shall show that an element in partial-derivativeCY(n, mu0, 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 X/S there is a conformally flat metric g of positive constant scalar curvature which is compatible with the distance d.