Preservation of nonnegative scalar curvature under convergence to a smooth limit

Determine whether Sormani–Wenger intrinsic flat convergence of a sequence of 3-dimensional Riemannian manifolds (M_j^3, g_j) with nonnegative scalar curvature to a smooth Riemannian manifold (M_0^3, g_0) implies that (M_0^3, g_0) also has nonnegative scalar curvature.

Background

The report discusses Gromov’s suggestion to formulate compactness for sequences with nonnegative scalar curvature and to develop a generalized notion of nonnegative scalar curvature on limits, with Sormani–Wenger intrinsic flat (SWIF) convergence as a candidate. While Gromov and Bamler proved preservation of nonnegative scalar curvature under C^0 convergence when all manifolds are diffeomorphic, SWIF enables consideration of sequences that are not diffeomorphic to the limit.

The authors note that tunneling constructions may yield counterexamples if tunneling is allowed, suggesting the issue is subtle for SWIF convergence without additional noncollapsing hypotheses.