Yamabe metrics of positive scalar curvature and conformally flat manifolds

Abstract

Let CY( n, μ, R 0 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]) ⩾μ 0 > 0, ∫ M|E g| n 2 dv g⩽R 0 , where [ g], μ( 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 ACY( n, μ 0, R 0 of CY( n, μ 0, R 0) in the space of all compact metric spaces equipped with the Hausdorff distance. We shall show that an element in ACY( n, μ 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 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.