On the Scalar Curvature Compactness Conjecture in the Conformal Case

Abstract

Is a sequence of Riemannian manifolds with positive scalar curvature, satisfying some conditions to keep the sequence reasonable, compact? What topology should one use for the convergence and what is the regularity of the limit space? In this paper we explore these questions by studying the case of a sequence of Riemannian manifolds which are conformal to the $n$-dimensional round sphere. We are able to show that the sequence of conformal factors are compact in several analytic senses and are able to establish $C^0$ convergence away from a singular set of small volume in a similar fashion as C. Dong. Under a bound on the total scalar curvature we are able to show that the limit conformal factor has weak positive scalar curvature in the sense of weakly solving the conformal positive scalar curvature equation.