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

TL;DR

This paper proves that sequences of three-dimensional warped product manifolds with nonnegative scalar curvature, volume bounds, and minimal surface area bounds have subsequences converging to a limit space with generalized nonnegative scalar curvature.

Contribution

It establishes pre-compactness and convergence of such manifolds to a limit space with a distributional nonnegative scalar curvature, extending Gromov and Sormani’s conjecture.

Findings

Subsequences converge to a $W^{1,p}$ Riemannian metric for all $p<2

Limit metric has nonnegative scalar curvature in the distributional sense

Sequence has a uniform upper volume bound and positive lower MinA bound

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.