Noncollapsed degeneration of Einstein
4–manifolds, II

Ozuch, Tristan

doi:10.2140/gt.2022.26.1529

Cited by 6 publications

(26 citation statements)

References 28 publications

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“…We finally recover the obstructions of [11] to the desingularization of Einstein metrics and many of the additional obstructions of [37] (but not the higher order obstructions of [39]). We conclude this section by proving that some Einstein orbifolds cannot be desingularized by smooth Einstein manifolds.…”

Section: Obstruction To the Desingularization Of Einstein Metricsmentioning

confidence: 87%

“…Let us now show that we recover the obstructions of [11,37] and find another expression for them. This new expression will further highlight the link between these obstructions and the lack of integrability of Einstein desingularizations of [38,Chapter 4].…”

Section: Obstructions To the Desingularizationmentioning

confidence: 95%

“…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%

“…satisfying 𝑔 𝑣 − (𝒈 𝑏 + 𝑣) ⟂ 𝐿 2 (𝒈 𝑏 ) 𝑶(𝒈 𝑏 ). We call Einstein modulo obstructions metrics such deformations which have been constructed in [37] (see also [32] in the smooth compact case, where a different vocabulary is used). We study the leading order of the obstruction along curves 𝑠 ↦ 𝑔 𝑠𝑣 for 𝑠 ∈ (−1, 1) at 𝑠 = 0.…”

Section: Desingularization Of Spherical and Hyperbolic Orbifoldsmentioning

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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Section: Obstruction To the Desingularization Of Einstein Metricsmentioning

confidence: 87%

Section: Obstructions To the Desingularizationmentioning

confidence: 95%

“…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%

Section: Desingularization Of Spherical and Hyperbolic Orbifoldsmentioning

confidence: 99%

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Integrability of Einstein deformations and desingularizations

Ozuch

2023

Comm Pure Appl Math

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“…If one could strengthen these estimates like in the recent work by Daniel Platt [61], then one should be able to deduce estimates like Kobayashi's using the Joyce construction. A third idea would be to build on more general theory of desingularizing Einstein orbifolds like the recent work by Ozuch [58,Theorem 4.6]. They deal with the problem of the injectivity radius being uncontrolled by introducing weights in their function spaces which counteract this.…”

Section: Some Alternative Proof Strategiesmentioning

confidence: 99%

Geodesics on a K3 surface near the orbifold limit

Olsen

2023

Ann Glob Anal Geom

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This article studies Kummer K3 surfaces close to the orbifold limit. We improve upon estimates for the Calabi–Yau metrics due to Kobayashi. As an application, we study stable closed geodesics. We use the metric estimates to show how there are generally restrictions on the existence of such geodesics. We also show how there can exist stable, closed geodesics in some highly symmetric circumstances due to hyperkähler identities.

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Analysis And Spectral Theory Of Neck-Stretching Problems

Langlais

2024

Commun. Math. Phys.

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We study the mapping properties of a large class of elliptic operators $$P_T$$ 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 $$T \rightarrow \infty $$ T → ∞ , we reduce the question of constructing approximate solutions of $$P_T u = f$$ 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 $$P_0$$ P 0 on the cylinder, we construct a Fredholm inverse for $$P_T$$ 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 $$G_2$$ G 2 -manifolds constructed by twisted connected sum. We relate our results to the swampland distance conjectures in physics.

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Noncollapsed degeneration of Einstein 4–manifolds, II

Cited by 6 publications

References 28 publications

Integrability of Einstein deformations and desingularizations

Integrability of Einstein deformations and desingularizations

Geodesics on a K3 surface near the orbifold limit

Analysis And Spectral Theory Of Neck-Stretching Problems

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