Convergence for Yamabe metrics of positive scalar curvature with integral bounds on curvature

Let Y-1(n, mu(0)) be the class of compact connected smooth n-manifolds M (n greater than or equal to 3) with Yamabe metrics g of unit volume which satisfy mu(M, [g]) greater than or equal to mu(0) > 0, where [g] and mu(M, [g]) denote the conformal class of g and the Yamabe invariant of (M, [g]), respectively. The purpose of this paper is to prove several convergence theorems for compact Riemannian manifolds in Y-1(n, mu(0)) with integral bounds on curvature. In particular, we present a pinching theorem for hat conformal structures of positive Yamabe invariant on compact 3-manifolds.