Preservation of nonnegative scalar curvature under SWIF with a MinA noncollapsing hypothesis

Determine whether, assuming the MinA(M_j)\ge A_0>0 noncollapsing condition, Sormani–Wenger intrinsic flat convergence of 3-dimensional Riemannian manifolds (M_j^3, g_j) with nonnegative scalar curvature to a smooth Riemannian manifold (M_0^3, g_0) ensures that (M_0^3, g_0) has nonnegative scalar curvature.

Background

The MinA hypothesis rules out collapsing via thin tunnels and other pathological behaviors that can invalidate scalar curvature properties in limits. The authors indicate that with MinA, tunneling-based counterexamples are prevented, suggesting MinA could be a sufficient noncollapsing condition for preserving nonnegative scalar curvature in the limit.

They also note that proving this in full generality would be challenging and propose examining the VADB convergence notion of Allen–Perales–Sormani as an easier case that implies SWIF convergence.

References

Open Question 2: If (M_j3,g_j) have Scalar \ge 0 and satisfy the MinA hypothesis and converge in the SWIF sense to smooth (M_0,g_0), does (M_0,g_0) have Scal\ge 0?

— 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 2

Sormani conjectured that a sequence of smooth manifolds with non-negative scalar curvature, uniform volume and diameter bounds, and a uniform minA lower bound (a type of non-collapsing condition) should converge in the volume preserving intrinsic flat sense to a limit space with non-negative scalar curvature.

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