Einstein Metrics, Four-Manifolds, and Differential Topology (Claude LeBrun)
Claude R. LeBrun, professeur à Stony Brook, spécialiste de géométrie riemannienne, est lauréat en 2025 d'une Chaire d'excellence de la FSMP, co-financée par l'IMJ-PRG. Il donnera dans le cadre de sa chaire 15h de cours sur le thème Einstein Metrics, Four-Manifolds, and Differential Topology.
Ce cours aura lieu à l'Institut Henri Poincaré (11 rue Pierre et Marie Curie, Paris 5e) du 19 mars au 23 avril 2026 (séances les jeudis de 14h à 17h15), en amphithéâtre Yvonne Choquet-Bruhat (bâtiment Perrin) pour les séances du 19 mars, 26 mars et 2 avril, puis en salle Pierre Grisvard (3e étage du bâtiment Borel) pour les autres séances.
Résumé du cours
These lectures will provide an overview of the Riemannian geometry of smooth compact 4-manifolds, with an emphasis on the interplay between differential topology and Riemannian geometry in dimension four. Our main focus will be on the problem of determining when a given 4-manifold can be “geometrized” by endowing it with an Einstein metric. There are actually various differential-topological obstructions that come in to play here, and which imply non-existence of Einstein metrics on many 4-manifolds; the fact that some of these delicately depend on the differentiable structure, and not just the homeomorphism type, will be illustrated through in-depth discussions of concrete examples. In the process, we will also familiarize ourselves with complementary collections of results and techniques that allow one to prove the existence of Einstein metrics on many interesting 4-manifolds. We will then carefully discuss the moduli spaces of Einstein metrics in some paradigmatic cases. When this moduli space is non-compact, we will also analyze the way that this failure of compactness arises through the bubbling-off of gravitational instantons, and/or collapse to lower-dimensional spaces.
In addition to discussing Einstein metrics, we will also discuss various other “canonical metric” problems that arise from interesting curvature functionals, primarily when these ideas play a supporting role in the theory of Einstein metrics.
Topics discussed will include:
The Seiberg-Witten equations; Yamabe invariants; scalar and Weyl curvature estimates; K¨ahler-Einstein metrics; conformally K¨ahler, Einstein metrics; Bach-flat metrics; rigidity theorems; moduli spaces of Einstein metrics; and gravitational instantons.
Pour en savoir plus sur les travaux de Claude LeBrun, vous pouvez consulter sa page personnelle.
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