Constructs uncountably many pairwise conformally inequivalent non-rotationally symmetric type II ancient Yamabe flows on S^n (n≥3) via non-radial inner-outer gluing after stereographic projection.
A dimension descent scheme for the positive mass theorem in arbitrary dimension
11 Pith papers cite this work. Polarity classification is still indexing.
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Pith papers citing it
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.
years
2026 11representative citing papers
Introduces a boundary analogue of the Gauss-Bonnet-Chern mass for asymptotically flat half-manifolds, proves it is well-defined, establishes positive mass theorems for graphical and conformally flat graphs, and provides a Penrose-type inequality.
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.
Overtorical manifolds and sufficiently regular enlargeable AM–PI spaces admit no positive (weighted) scalar curvature, proved by singular dimension descent with conformal blow-ups.
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
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.
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.
Defines a mass-type invariant for asymptotically Schwarzschild smooth metric measure spaces modeled on the Euclidean half-space and relates it to the fractional Yamabe problem.
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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Uncountably many non-rotationally symmetric type II ancient Yamabe flows on the sphere
Constructs uncountably many pairwise conformally inequivalent non-rotationally symmetric type II ancient Yamabe flows on S^n (n≥3) via non-radial inner-outer gluing after stereographic projection.
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A new boundary mass for asymptotically flat half-manifolds
Introduces a boundary analogue of the Gauss-Bonnet-Chern mass for asymptotically flat half-manifolds, proves it is well-defined, establishes positive mass theorems for graphical and conformally flat graphs, and provides a Penrose-type inequality.
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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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Positive Scalar Curvature Obstructions via Singular Dimension Descent
Overtorical manifolds and sufficiently regular enlargeable AM–PI spaces admit no positive (weighted) scalar curvature, proved by singular dimension descent with conformal blow-ups.
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Riemannian Penrose inequality in all dimensions
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
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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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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
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A mass-type invariant for smooth metric measure spaces and its relation with the fractional Yamabe problem
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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