The Riemannian Penrose inequality is proven in arbitrary dimensions for smooth complete asymptotically flat manifolds with nonnegative scalar curvature and compact outer-minimizing minimal boundary allowing singular sets of Hausdorff dimension at most n-8, with equality only for Riemannian Schwarzs
A dimension descent scheme for the positive mass theorem in arbitrary dimension
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abstract
We describe how the Schoen-Yau proof of the positive mass theorem can be extended to arbitrary dimensions. To overcome the problem of singularities, we propose a new inductive scheme. To carry out the inductive step, we use a combination of several techniques, including the shielding principle of Lesourd-Unger-Yau, as well as a conformal blow-up argument in the spirit of Bi-Hao-He-Shi-Zhu. Our arguments also rely on the Cheeger-Naber bound for the Minkowski dimension of the singular set.
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math.DG 4years
2026 4verdicts
UNVERDICTED 4representative citing papers
The positive mass theorem holds for complete asymptotically hyperbolic manifolds satisfying the dominant energy condition, including those with arbitrary ends.
Proves the spacetime positive mass theorem for asymptotically flat and asymptotically hyperboloidal initial data sets in arbitrary dimensions using Brendle-Wang's Riemannian positive mass theorem.
The spacetime positive energy theorem in dimensions n ≥ 4 is obtained by reducing it to the Riemannian positive mass theorem using the Jang equation with a capillary term.
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Positive mass theorem for initial data sets with arbitrary ends
The positive mass theorem holds for complete asymptotically hyperbolic manifolds satisfying the dominant energy condition, including those with arbitrary ends.
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The Hyperboloidal and Spacetime Positive Mass Theorem in All Dimensions
Proves the spacetime positive mass theorem for asymptotically flat and asymptotically hyperboloidal initial data sets in arbitrary dimensions using Brendle-Wang's Riemannian positive mass theorem.
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On the spacetime positive energy theorem in arbitrary dimension
The spacetime positive energy theorem in dimensions n ≥ 4 is obtained by reducing it to the Riemannian positive mass theorem using the Jang equation with a capillary term.