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Scalar curvature rigidity of spheres with subsets removed and $L^\infty$ metrics

· 2024 · math.DG · arXiv 2407.21312

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abstract

We prove the scalar curvature rigidity for $L^\infty$ metrics on $\mathbb S^n\backslash\Sigma$, where $\mathbb S^n$ is the $n$-dimensional sphere with $n\geq 3$ and $\Sigma$ is a closed subset of $\mathbb S^n$ of codimension at least $\frac{n}{2}+1$ that satisfies the wrapping property. The notion of wrapping property was introduced by the second author for studying related scalar curvature rigidity problems on spheres. For example, any closed subset of $\mathbb S^n$ contained in a hemisphere and any finite subset of $\mathbb S^n$ satisfy the wrapping property. The same techniques also apply to prove an analogous scalar rigidity result for $L^\infty$ metrics on tori that are smooth away from certain subsets of codimension at least $\frac{n}{2}+1$. As a corollary, we obtain a positive mass theorem for complete asymptotically flat spin manifolds with arbitrary ends for $L^\infty$ metrics.

fields

math.DG 2

years

2026 2

verdicts

ACCEPT 1 UNVERDICTED 1

representative citing papers

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

math.DG · 2026-06-19 · unverdicted · novelty 5.0

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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Showing 2 of 2 citing papers.

  • Riemannian Positive Mass Theorem in All Dimensions in the Presence of Low-Codimension Singularities math.DG · 2026-06-22 · accept · none · ref 52 · internal anchor

    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.

  • $L^\infty$-metrics on tori and Schoen's conjecture math.DG · 2026-06-19 · unverdicted · none · ref 22 · internal anchor

    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