Gromov–Sormani MinA scalar compactness conjecture

Let $\{M_i^3\}$ be a sequence of closed oriented three-dimensional Riemannian manifolds without boundary. Write $\operatorname{Scal}_{g_i}$ for scalar curvature, $\operatorname{Vol}_{g_i}$ for volume, $\operatorname{Diam}_{g_i}$ for diameter, and

$$\operatorname{MinA}(M_i^3,g_i):=\inf\{\operatorname{Area}(\Sigma)\mid \Sigma\text{ is a closed minimal surface in }M_i^3\}.$$

Assume that, for constants $V,D,A_0>0$,

$$\operatorname{Scal}_{g_i}\geq 0,\qquad \operatorname{Vol}_{g_i}(M_i)\leq V,\qquad \operatorname{Diam}_{g_i}(M_i)\leq D,$$

and

$$\operatorname{MinA}(M_i^3,g_i)\geq A_0.$$

Gromov–Sormani MinA scalar compactness conjecture. A subsequence converges in the volume-preserving intrinsic flat sense to a three-dimensional rectifiable limit space $M_\infty$. Furthermore, $M_\infty$ is expected to be a connected geodesic metric space with Euclidean tangent cones almost everywhere and with a generalized notion of nonnegative scalar curvature.

The conjecture is a scalar-curvature compactness statement designed to rule out collapsing phenomena through the positive lower bound on the areas of closed minimal surfaces. It was verified in the rotationally symmetric setting by Park, Tian and Wang, while the present paper verifies it for the stated warped-product class.