Differential Geometry Seminar: Noncollapsed degeneration of Einstein 4-manifolds

Seminar | December 9 | 3-4 p.m. | 939 Evans Hall

 Tristan Ozuch-Meersseman, ENS

 Department of Mathematics

We study unit-volume Einstein 4-manifolds with bounded diameter and their possible degenerations. These degenerations were understood in the 80's by Anderson and Bando-Kasue-Nakajima in a Gromov-Hausdorff sense. We first obtain a finer convergence in weighted Hölder spaces and use it to prove that any smooth Einstein 4-manifold close to a singular one in a mere Gromov-Hausdorff sense is the result of a gluing-perturbation procedure that we develop. This new description lets us extend Biquard's obstruction to a general setting, allowing multiple singularities and trees of singularities, and only assuming a Gromov-Hausdorff convergence. It also sheds light on the structure of the moduli space of Einstein 4-manifolds near its boundary.