$L^{\mathrm{\infty }}$-metrics on tori and Schoen's conjecture

We prove Schoen’s conjecture on $L^{\mathrm{\infty }}$-metrics for tori under an additional assumption on the fundamental group of the singular set. More precisely, we consider an $L^{\mathrm{\infty }}$-metric on a torus that is smooth and has non-negative scalar curvature away from a singular set of Minkowski dimension at most $n-3+(n-1{)}^{-1}$. We show that if the induced homomorphism from the fundamental group of the singular set to the fundamental group of the torus is not surjective, then this metric extends to a smooth flat metric on the torus. Our proof uses weighted scalar curvature and the relative index theorem.