On the Scalar Curvature Compactness Conjecture in the Conformal Case

TL;DR

This paper investigates the compactness properties of sequences of conformally related Riemannian manifolds with positive scalar curvature, establishing convergence results and analyzing the regularity and curvature of the limit space.

Contribution

It demonstrates the compactness of conformal factors in several analytic senses and characterizes the limit's scalar curvature under bounded total scalar curvature.

Findings

Conformal factors are compact in multiple analytic senses.

Established $C^0$ convergence away from a small-volume singular set.

Limit conformal factor has weak positive scalar curvature.

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.