# Geometric Analysis and Topology Seminar

#### Noncollapsed degeneration of Einstein 4-manifolds

**Speaker:**
Tristan Ozuch-Meersseman, Ecole Normale Superieure

**Location:**
Warren Weaver Hall 512

**Date:**
Friday, November 15, 2019, 11 a.m.

**Synopsis:**

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.