Noncollapsed degeneration of Einstein 4-manifolds I

Ozuch, Tristan

doi:10.48550/arxiv.1909.12957

Cited by 2 publications

(7 citation statements)

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“…Examples of Einstein desingularizations of sine cones due to Böhm [6] suggest that it can also happen that there is only a discrete sequence approaching the singular metric rather than a smooth curve. By analogy with recent results of Ozuch [26,27,29], we believe that the obstructions we found also exclude such desingularizations and we therefore propose the following two conjectures.…”

Section: Theorem Dsupporting

confidence: 87%

Topology of asymptotically conical Calabi--Yau and G2 manifolds and desingularization of nearly Kähler and nearly G2 conifolds

Schiemanowski¹

2022

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A natural approach to the construction of nearly G2 manifolds lies in resolving nearly G2 spaces with isolated conical singularities by gluing in asymptotically conical G2 manifolds modelled on the same cone. If such a resolution exits, one expects there to be a family of nearly G2 manifolds, whose endpoint is the original nearly G2 conifold and whose parameter is the scale of the glued in asymptotically conical G2 manifold. We show that in many cases such a curve does not exist.The non-existence result is based on a topological result for asymptotically conical G2 manifolds: if the rate of the metric is below −7/2, then the G2 4-form is exact if and only if the manifold is Euclidean R 7 .A similar construction is possible in the nearly Kähler case, which we investigate in the same manner with similar results. In this case, the non-existence results is based on a topological result for asymptotically conical Calabi-Yau 6-manifolds: if the rate of the metric is below −3, then the square of the Kähler form and the complex volume form can only be simultaneously exact, if the manifold is Euclidean R 6 .

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Section: Theorem Dsupporting

confidence: 87%

Topology of asymptotically conical Calabi--Yau and G2 manifolds and desingularization of nearly Kähler and nearly G2 conifolds

Schiemanowski¹

2022

Preprint

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“…Any smooth Einstein 4-manifold close to a compact Einstein orbifold in a mere Gromov-Hausdorff sense has been recently been produced by a gluing-perturbation procedure [Ozu19a,Ozu19b]. Here, this lets us understand the obstructions to the d GH -desingularization of an Einstein orbifold through obstructions similar to (5) and (7).…”

Section: Question 03 Can We Find Similar Obstructions For Einstein De...mentioning

confidence: 99%

“…Let us assume that there exists a sequence of Einstein metrics (M, g n ) n converging to an Einstein orbifold (M o , g o ) with E(g o ) + λg o = 0. Then, according to [Ozu19a,Ozu19b,Ozu20], up to taking a subsequence, there exist…”

Section: Obstruction To the Desingularization Of Spherical And Hyperb...mentioning

confidence: 99%

“…Higher dimensional Einstein orbifolds with isolated singularities. The work of [Ozu19a,Ozu19b] extends almost verbatim to the degeneration of Einstein d-manifolds (d 5) satisfying the (non natural) assumption that the L d 2 -norm of its curvature is bounded. It shows that, exactly like in dimension 4, the possible Gromov-Hausdorff limits are Einstein orbifolds with isolated singularities and the singularity models are Ricci-flat ALE orbifolds.…”

Section: Obstruction To the Desingularization Of Spherical And Hyperb...mentioning

confidence: 99%

“…It shows that, exactly like in dimension 4, the possible Gromov-Hausdorff limits are Einstein orbifolds with isolated singularities and the singularity models are Ricci-flat ALE orbifolds. Indeed the results of [Ozu19a,Ozu19b,Ozu20t] only use the fact that the dimension is 4 to obtain a bound on the L 2 -norm of the Riemannian curvature from the noncollapsedness assumption.…”

Section: Obstruction To the Desingularization Of Spherical And Hyperb...mentioning

confidence: 99%

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Integrability of Einstein deformations and desingularizations

Ozuch¹

2021

Preprint

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We study the question of the integrability of Einstein deformations and relate it to the question of the desingularization of Einstein metrics. Our main application is a negative answer to the long-standing question of whether or not every Einstein 4-orbifold (which is an Einstein metric space in a synthetic sense) is limit of smooth Einstein 4-manifolds. We more precisely show that spherical and hyperbolic 4-orbifolds with the simplest singularities cannot be Gromov-Hausdorff limits of smooth Einstein 4metrics without relying on previous integrability assumptions. For this, we analyze the integrability of deformations of Ricci-flat ALE metrics through variations of Schoen's Pohozaev identity. Inspired by Taub's preserved quantity in General Relativity, we also introduce preserved integral quantities based on the symmetries of Einstein metrics. These quantities are obstructions to the integrability of infinitesimal Einstein deformations "closing up" inside a hypersurface -even with change of topology. We show that many previously identified obstructions to the desingularization of Einstein 4-metrics are equivalent to these quantities on Ricci-flat cones. In particular, all of the obstructions to desingularizations bubbling out Eguchi-Hanson metrics are recovered. This lets us further interpret the obstructions to the desingularization of Einstein metrics as a defect of integrability. Contents INTEGRABILITY OF EINSTEIN DEFORMATIONS AND DESINGULARIZATIONSA.1. A local gauge for Einstein metrics 30 A.2. Vanishing of the obstructions to Einstein deformations 31 Appendix B. Function spaces and analyticity on ALE spaces 32 B.1. Function spaces 32 B.2. Real-analytic dependence of Einstein modulo obstructions metrics 32 References 33

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Noncollapsed degeneration of Einstein 4-manifolds I

Cited by 2 publications

References 0 publications

Topology of asymptotically conical Calabi--Yau and G2 manifolds and desingularization of nearly Kähler and nearly G2 conifolds

Topology of asymptotically conical Calabi--Yau and G2 manifolds and desingularization of nearly Kähler and nearly G2 conifolds

Integrability of Einstein deformations and desingularizations

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