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Positive scalar curvature and isolated conical singularity
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abstract
We prove a Geroch type result for isolated conical singularity. Namely, we show that there is no Riemannian metric $g$ on $ X \# T^n $ with an isolated conical singularity which has nonnegative scalar curvature on the regular part, and is positive at some point. In particular, this implies that there is no metric on tori with an isolated conical singularity and positive scalar curvature. We also prove that a scalar flat Riemannian metric $g$ on $X \# T^n$ with finitely many isolated conical singularities must be flat, and extend smoothly across the singular points. We do not a priori assume that a conically singular point on $X$ is a manifold point; i.e., the cross section of the conical singularity may not be spherical.
Forward citations
Cited by 2 Pith papers
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Riemannian Positive Mass Theorem in All Dimensions in the Presence of Low-Codimension Singularities
Proves Riemannian positive mass theorem for asymptotically flat L^∞ metrics with subcritical singular sets of Minkowski dimension less than n-3 + 2/n (rigidity for ≤ n-3 + 1/(n-1)), using density theorem, capacity est...
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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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