Singular Metrics with Nonnegative Scalar Curvature and RCD

We show that a uniformly Euclidean metric with isolated singularity on closed [Formula: see text], where [Formula: see text] or [Formula: see text], [Formula: see text] spin, and nonnegative scalar curvature on the smooth part is flat and extends smoothly over the singularity. This confirms Schoen’s Conjecture in these cases. The novel approach here, which is the key to the proof, is to show that the space has nonnegative synthetic Ricci curvature, i.e., an [Formula: see text] space. Our result also holds when the singular set consists of a finite union of submanifolds (of possibly different dimensions) intersecting transversally under additional assumption on the co-dimension and the location of the singular set.