Yamabe metrics of positive scalar curvature and conformally flat manifolds

Abstract

Let 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

$$\text{μ(M,[g]) ⩾μ}{}_0\text{> 0, ∫}{}_{\mathrm{M}}\text{|E}{}_{\mathrm{g}}\text{|}\text{n}\text{2}\text{dv}{}_{\mathrm{g}}\text{⩽R}{}_0$$

, 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, 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 $\text{F}\text{S}$ there is a conformally flat metric g of positive constant scalar curvature which is compatible with the distance d.