Einstein Metrics, Four-Manifolds, and Differential Topology (Claude LeBrun)
Claude R. LeBrun, professor at Stony Brook, specialist in Riemannian Geometry, is laureate of a FSMP Chair of Excellence, co-funded by IMJ-PRG, in 2025. He is giving, in the frame of his chair, a 15-hour course on the topic Einstein Metrics, Four-Manifolds, and Differential Topology.
The lectures will take place at the IHP from March 19th to April 16th 2026 (Thursdays from 2:00 p.m. to 5:15 p.m.). The lectures of March 19th, March 26th and April 2nd will take place in amphitheatre Yvonne Choquet-Bruhat (Perrin building), and the others in room Pierre Grisvard (3rd floor of the Borel building).
Abstract
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.
Read more about Claude LeBrun's work on his personal web page.
Lecture 1 :
Download here the slides of the first lecture.
Lecture 2 :
Download here the slides of the second lecture.
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