Gromov and Sormani conjectured that under suitable geometric uniform bounds, sequences of closed 3-dimensional Riemannian manifolds with nonnegative scalar curvature converge in a certain sense.

Specifically, let {(M_j^3, g_j)}_{j=1}^\infty be a sequence of compact 3-dimensional orientable closed Riemannian manifold. Assume that each (M_j^3,g_j) has non-negative scalar curvature. Also, assume that the volume and diameter are uniformly bounded above and that the area of closed minimal surfaces is uniformly bounded away from zero (denoted as the minA lower bound). Then, the conjecture states that there exists a subsequence that converges in the intrinsic flat sense to the limit space, which has nonnegative scalar curvature in the generalized sense.

In this talk, we discuss two special cases. One case is the sequence of rotationally symmetric 3-manifold. In this case, we confirm the conjecture. The other case is the sequence of warped product S^2\times S^1. In this case we discuss an example that shows the sharp regularity for the limit space. We also prove an analytic convergence for this case. The proof combines the minA lower bound with the spherical means inequality and gives a new way of applying the minA lower bound.