Schoen’s conjecture on scalar-curvature singularities

Schoen’s conjecture links removal of scalar-curvature singularities with rigidity of Einstein metrics. The available literature now contains both claimed high-dimensional counterexamples to the unrestricted statement and positive results under stronger regularity and codimension assumptions.

Known results

• Li and Mantoulidis (2016): partial rigidity for continuous metrics with $W^{1,p}_{\mathrm{loc}}$ regularity, $p>n$, and sufficiently large singular-set codimension; extra Lipschitz regularity yields smooth Einstein metrics.
• Li and Mantoulidis (2021): continuous metrics with singular sets of codimension $>2$ extend, up to a bi-Lipschitz change of coordinates, to smooth Einstein metrics.
• Cecchini, Frenck, and Zeidler (2024): for every $n\ge 8$, claimed $L^\infty$ counterexamples exist, including isolated singularities in arbitrarily high dimensions.
• A 2026 torus result gives smooth extension under additional Minkowski-dimension and fundamental-group hypotheses.

September 2026 restricted proof claim

The latest paper claims further extension and smooth-removability results for continuous metrics and for $L^\infty$ metrics sufficiently close to a smooth background, under explicit codimension and regularity assumptions. It does not establish the unrestricted conjecture.

Current status (as of September 2026): the unrestricted conjecture is claimed refuted in dimensions $n\ge 8$, while several restricted continuous or $L^\infty$ variants have claimed positive results and the general problem remains open.