Scalar MinA Compactness for Warped Product Manifolds

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
precompactness of a sequence of three dimensional warped product manifolds with warped
circles over standard spheres 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 W1,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. This is joint work with Wenchuan Tian (arxiv to
appear this week). See also previous work with W. Tian and Jiewon Park
(https://arxiv.org/abs/1812.03502).