Compactness of sequences of warped product circles over spheres with nonnegative scalar curvature

Abstract

Gromov and Sormani conjectured that a sequence of three dimensional Riemannian manifolds with nonnegative scalar curvature and some additional uniform geometric bounds should have a subsequence which converges in some sense to a limit space with generalized notion of nonnegative scalar curvature. In this paper, we study the pre-compactness of a sequence of three dimensional warped product manifolds with warped circles over standard ${\mathbb{S}}^2$ that have nonnegative scalar curvature, a uniform upper bound on the volume, and a positive uniform lower bound on the MinA, which is the minimum area of closed minimal surfaces in the manifold. We prove that such a sequence has a subsequence converging to a $W^{1,p}$ Riemannian metric for all $p<2$, and that the limit metric has nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch.

add references and related discussions in Introduction, add a subsection for uniform systole lower estimate, add a subsection for reviewing some basic notions and results about Min-Max minimal surface theory that we need in this paper, 56 pages, submitted version