Noncollapsed degeneration of Einstein
4–manifolds, I

Ozuch, Tristan

doi:10.2140/gt.2022.26.1483

Cited by 3 publications

(12 citation statements)

References 18 publications

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“…This allows us to use weighted analysis to study the equations P i u i = f i in a range where the operators P i satisfy the Fredholm property. Similar ideas can be found for instance in [35].…”

Section: Results and Strategysupporting

confidence: 76%

Analysis and spectral theory of neck-stretching problems

Langlais¹

2023

Preprint

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We study the mapping properties of a large class of elliptic operators P T in gluing problems where two non-compact manifolds with asymptotically cylindrical geometry are glued along a neck of length 2T . In the limit where T → ∞, we reduce the question of constructing approximate solutions of P T u = f to a finite-dimensional linear system, and provide a geometric interpretation of the obstructions to solving this system. Under some assumptions on the real roots of the model operator P 0 on the cylinder, we are able to construct a Fredholm inverse for P T with good control on the growth of its norm. As applications of our method, we study the decay rate and density of the low eigenvalues of the Laplacian acting on differential forms, and give improved estimates for compact G 2 -manifolds constructed by twisted connected sum. We relate our results to the swampland distance conjectures in physics.

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Section: Results and Strategysupporting

confidence: 76%

Analysis and spectral theory of neck-stretching problems

Langlais¹

2023

Preprint

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“…𝒆 (𝐻 2 , 𝐻 4 ) = 0 because 𝐻 4 is anti-selfdual, and we recover the obstruction (36) in this situation. The other obstructions with Killing vector fields are recovered in the same way.…”

Section: Einstein Metrics Closing-up Inside a Hypersurfacesupporting

confidence: 60%

“…Since both ∑ 𝑖 ℎ + 𝑖𝑖 = 0 and ∑ 𝑘 ℎ − 𝑘𝑘 = 0, the obstruction ( 37) is satisfied. The proof is similar for the other obstructions (36). □…”

Section: Development Of Ricci-flat Ale Metricsmentioning

confidence: 66%

“…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 [36,37].…”

Section: Obstruction To the Desingularization Of Einstein 4-manifoldsmentioning

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 [36][37][38] 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: Higher Dimensional Einstein Orbifolds With Isolated Singular...mentioning

confidence: 99%

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Integrability of Einstein deformations and desingularizations

Ozuch

2023

Comm Pure Appl Math

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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 4‐metrics 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 conserved quantity in General Relativity, we also introduce conserved 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 off Eguchi‐Hanson metrics are recovered. This lets us further interpret the obstructions to the desingularization of Einstein metrics as a defect of integrability.

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