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 C0 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.

References

Open Question 1: Suppose a sequence, (M_j3, g_j), with Scal\ge 0 converge in some sense to a smooth Riemannian manifold, (M_03, g_0). Does (M_03, g_0) have Scal\ge 0? SWIF convergence allows one to study sequences which are not diffeomorphic. Does this work for SWIF convergence?

— Oberwolfach Workshop Report: Analysis, Geometry and Topology of Positive Scalar Curvature Metrcs: Limits of sequences of manifolds with nonnegative scalar curvature and other hypotheses  (2404.17121 - Tian et al., 2024) in Main text, Open Question 1

Assume that smooth manifolds $(M_k,g_k)$ with non-negative scalar curvature converge (weakly) to a smooth limit $(M,g)$. Which notions of weak convergence ensure that the limit space also has non-negative scalar curvature?

— Scalar Curvature Lower Bounds and Convergence in Measure  (2609.34141 - Mazurowski et al., 28 Sep 2026) in Question 1.3, Section 1, subsection “Discussion”

Fix any number $1 \le p < \infty$. Assume that smooth metrics $g_k$ and $g$ satisfy $g_k \to g$ in $Lp$ and $g_k{-1}\to g_k{-1}$ in $Lp$. If each $g_k$ has non-negative scalar curvature, does it follow that $g$ has non-negative scalar curvature? If this is not true, is there any non-collapsing condition weaker than $\nu(g_k,2)\ge -\delta$ under which it is true?

— Scalar Curvature Lower Bounds and Convergence in Measure  (2609.34141 - Mazurowski et al., 28 Sep 2026) in Question 1.4, Section 1, subsection “Discussion”