Regularity of Einstein 5-manifolds via 4-dimensional gap theorems
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914332593356800 |
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| author | Huang, Yiqi Ozuch, Tristan |
| author_facet | Huang, Yiqi Ozuch, Tristan |
| contents | We refine the regularity of noncollapsed limits of 5-dimensional manifolds with bounded Ricci curvature. In particular, for noncollapsed limits of Einstein 5-manifolds, we prove that
(1) tangent cones are unique of the form $\mathbb{R}\times\mathbb{R}^4/Γ$ on the top stratum, hence outside a countable set of points; this follows from a new isolation result for cones of the form $\mathbb{R}\times\mathbb{R}^4/Γ$ among all tangent cones,
(2) the singular set is entirely contained in a countable union of Lipschitz curves and points,
(3) away from a nowhere dense subset, these Lipschitz curves consist of smooth geodesics,
(4) the interior of any geodesic is removable: limits of Einstein manifolds are real-analytic orbifolds with singularities along geodesic and bounded curvature away from their extreme points, and
(5) if an asymptotically Ricci-flat 5-manifold with Euclidean volume growth has one tangent cone at infinity that splits off a line, then it is the unique tangent cone at infinity.
These results prompt the question of the orbifold regularity of noncollapsed limits of Einstein manifolds off a codimension 5 set in arbitrary dimension.
The proofs rely on a new result of independent interest: all spherical and hyperbolic 4-orbifolds are isolated among Einstein 4-orbifolds in the Gromov-Hausdorff sense. This yields various gap theorems for Einstein 4-orbifolds, which do not extend to higher dimensions. The proofs of these gap theorems require a careful analysis of singular metrics and families of metrics that degenerate. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21317 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity of Einstein 5-manifolds via 4-dimensional gap theorems Huang, Yiqi Ozuch, Tristan Differential Geometry Analysis of PDEs Metric Geometry We refine the regularity of noncollapsed limits of 5-dimensional manifolds with bounded Ricci curvature. In particular, for noncollapsed limits of Einstein 5-manifolds, we prove that (1) tangent cones are unique of the form $\mathbb{R}\times\mathbb{R}^4/Γ$ on the top stratum, hence outside a countable set of points; this follows from a new isolation result for cones of the form $\mathbb{R}\times\mathbb{R}^4/Γ$ among all tangent cones, (2) the singular set is entirely contained in a countable union of Lipschitz curves and points, (3) away from a nowhere dense subset, these Lipschitz curves consist of smooth geodesics, (4) the interior of any geodesic is removable: limits of Einstein manifolds are real-analytic orbifolds with singularities along geodesic and bounded curvature away from their extreme points, and (5) if an asymptotically Ricci-flat 5-manifold with Euclidean volume growth has one tangent cone at infinity that splits off a line, then it is the unique tangent cone at infinity. These results prompt the question of the orbifold regularity of noncollapsed limits of Einstein manifolds off a codimension 5 set in arbitrary dimension. The proofs rely on a new result of independent interest: all spherical and hyperbolic 4-orbifolds are isolated among Einstein 4-orbifolds in the Gromov-Hausdorff sense. This yields various gap theorems for Einstein 4-orbifolds, which do not extend to higher dimensions. The proofs of these gap theorems require a careful analysis of singular metrics and families of metrics that degenerate. |
| title | Regularity of Einstein 5-manifolds via 4-dimensional gap theorems |
| topic | Differential Geometry Analysis of PDEs Metric Geometry |
| url | https://arxiv.org/abs/2512.21317 |