Title: On the Scalar Curvature Compactness Conjecture in the Conformal Case Show Chinese title

Title: 关于共形情形下的标量曲率紧致性猜想

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. Show Chinese abstract

Abstract: 是一个具有正标量曲率的黎曼流形序列，在满足某些条件以保持序列合理的情况下，是否紧致？ 应该使用哪种拓扑来进行收敛，极限空间的正则性如何？ 在本文中，我们通过研究一个与$n$维球面共形的黎曼流形序列来探讨这些问题。 我们能够证明共形因子序列在几种分析意义上是紧致的，并能够像 C. Dong 一样，在一个小体积的奇异集之外建立$C^0$收敛。 在总标量曲率有界的情况下，我们能够证明极限共形因子在弱求解共形正标量曲率方程的意义下具有弱正标量曲率。