An Extreme Example related to the Compactness Conjecture for Scalar ge 0 and MinA ge A

In 2014, Gromov vaguely conjectured that a sequence of manifolds with nonnegative scalar
curvature should have a subsequence which converges in some weak sense to a limit space
with some generalized notion of nonnegative scalar curvature. The conjecture has been
made precise at an IAS Emerging Topics meeting: requiring that the sequence be three
dimensional with uniform upper bounds on diameter and volume, and a positive uniform
lower bound on MinA, which is the minimum area of a closed minimal surface in the manifold
(https://arxiv.org/abs/2103.10093). Here we present a sequence of warped product
manifolds with warped circles over standard spheres, that have circular fibres over the poles
whose length diverges to infinity, that satisfy the hypotheses of this IAS conjecture. We
prove this sequence converges in the W1,p sense for p<2 an extreme limit space that has
nonnegative scalar curvature in the distributional sense as defined by Lee-LeFloch and that
the total distributional scalar curvature converges. The sequences also converges in the
uniform, Gromov-Hausdorff, and Intrinsic Flat sense to the metric completion of the extreme
limit space which is homeomorphic to a product of a standard two sphere with a circle. This
is joint work with Changliang Wang and Christina Sormani (https://arxiv.org/abs/2304.07000)
and a second paper to appear with C. Sormani.