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Tristan Ozuch: Lectures

May 27, 2021
TITLE: Noncollapsed degeneration and desingularization of Einstein 4-manifolds

We study the noncollapsed singularity formation of Einstein 4-manifolds. We prove that any smooth Einstein 4-manifold close to a singular one in a mere Gromov-Hausdorff (GH) sense is the result of a gluing-perturbation procedure that we develop and which handles the presence of multiple trees of singularities at arbitrary scales.

This sheds light on the structure of the moduli space of Einstein 4-manifolds near its boundary and lets us show that spherical and hyperbolic orbifolds cannot be GH-approximated by smooth Einstein metrics. New obstructions specific to the compact situation moreover raise the question of whether or not a sequence of Einstein 4-manifolds degenerating while bubbling out gravitational instantons has to be Kähler-Einstein. These obstructions are even more restrictive in the Ricci-flat situation.

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