Schoen’s conjecture on scalar-curvature singularities

Prove Schoen’s conjecture that if an L-infinity Riemannian manifold has nonnegative scalar curvature away from a smooth, closed, embedded singular submanifold of codimension at least three and has non-positive Yamabe invariant, then the metric extends smoothly across the singular set as a Ricci-flat metric.

Background

The paper formulates a conjecture attributed to Schoen concerning the removability of scalar-curvature singularities. It concerns an L-infinity Riemannian metric on a manifold whose scalar curvature is nonnegative away from a smooth singular submanifold of codimension at least three, under the additional assumption that the manifold has non-positive Yamabe invariant.

The authors state that the conjecture is known in dimension three, that partial results exist in higher dimensions, and that it fails in general in sufficiently high dimensions. They decompose the conjecture into establishing Ricci-flatness on the nonsingular region and then proving smooth extension across the singular set. The paper addresses the extension step under its hypotheses, but does not resolve the full conjecture in general.

References

As we will explain below, our results can be applied to the following conjecture due to Schoen; see, e.g., Li--MantoulidisConjecture 1.5. Suppose that $(Mn, g)$ is an $L\infty$-Riemannian manifold, and $S\subset M$ is a smooth closed embedded submanifold of codimension at least 3. Suppose that $g$ has non-negative scalar curvature on $M\setminus S$, and $M$ has non-positive Yamabe invariant. Then $g$ extends smoothly to $M$ as a metric with zero Ricci curvature.

— Removability of non-isolated singularities for Einstein metrics and RCD spaces  (2609.01464 - Antonelli et al., 1 Sep 2026) in Conjecture 1.4 (Introduction)