Asymptotically flat L^∞ ∩ W^{1,n} manifolds with non-negative distributional scalar curvature satisfy the positive mass theorem, with zero mass implying global isometry to Euclidean space via an integral distance.
Positive scalar curvature with point singularities
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ADM mass is nonnegative (and zero only for Euclidean space under a slightly stronger dimension bound) for complete AF L∞ metrics with nonnegative scalar curvature outside a singular set of Minkowski dimension < n−3+2/n.
Under a non-surjectivity assumption on the fundamental group homomorphism from the singular set, an L^∞ metric on a torus with non-negative scalar curvature outside a Minkowski dimension ≤ n-3+(n-1)^{-1} singular set extends to a smooth flat metric, proved via weighted scalar curvature and the relat
Proves scalar curvature rigidity for L^∞ metrics on S^n minus high-codimension subsets with wrapping property, plus analogous result for tori and positive mass theorem corollary for L^∞ AF spin manifolds.
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Spaces with distributional scalar curvature bounded from below: Optimal regularity and positive mass
Asymptotically flat L^∞ ∩ W^{1,n} manifolds with non-negative distributional scalar curvature satisfy the positive mass theorem, with zero mass implying global isometry to Euclidean space via an integral distance.
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Riemannian Positive Mass Theorem in All Dimensions in the Presence of Low-Codimension Singularities
ADM mass is nonnegative (and zero only for Euclidean space under a slightly stronger dimension bound) for complete AF L∞ metrics with nonnegative scalar curvature outside a singular set of Minkowski dimension < n−3+2/n.
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$L^\infty$-metrics on tori and Schoen's conjecture
Under a non-surjectivity assumption on the fundamental group homomorphism from the singular set, an L^∞ metric on a torus with non-negative scalar curvature outside a Minkowski dimension ≤ n-3+(n-1)^{-1} singular set extends to a smooth flat metric, proved via weighted scalar curvature and the relat
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Scalar curvature rigidity of spheres with subsets removed and $L^\infty$ metrics
Proves scalar curvature rigidity for L^∞ metrics on S^n minus high-codimension subsets with wrapping property, plus analogous result for tori and positive mass theorem corollary for L^∞ AF spin manifolds.