2023/07/26 by Tristan Ozuch · 2 citations
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Einstein #Einstein's constant #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Gravitational singularity #Homogeneous space #Hypersurface #Limit (mathematics) #Mathematical analysis #Mathematical physics #Mathematics #Orbifold #Physics #Pure mathematics #Theoretical physics
paper · pdf · doi:10.1002/cpa.22129
published in Communications on Pure and Applied Mathematics 77(1), 177-220 (Wiley)
openalex publication_date 2023/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25
Abstract 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.