Colloquium

Claude LeBrun, Stony Brook University

"Four-Manifolds, Conformal Curvature, and Differential Topology"

Gauss discovered that any Riemannian 2-manifold is locally conformally flat, in the sense that, near any point, there is a coordinate system in which the metric becomes a positive function times the Euclidean metric. However, this paradigm generally fails for Riemannian manifolds of higher dimension; in other words, most higher-dimensional manifolds are not “locally conformally flat”. Indeed, when the dimension is at least 4, Weyl discovered that a piece of the Riemann curvature tensor, now known as the “Weyl tensor”, vanishes identically if and only if the given metric is locally conformally flat. Given a fixed smooth compact 4-manifold without boundary, the “Weyl functional” is by definition the L2-norm-squared of the Weyl tensor, considered as a non-negative function on the space of Riemannian metrics. Its infimum over all metrics then provides a fascinating differential-topological invariant of the given smooth, compact 4-manifold. It turns out that there are many 4-manifolds for which this invariant can be exactly calculated, and there are even large classes of manifolds on which the infimum is achieved. However, our current understanding of this problem remains distinctly limited. In this talk, I will explain some recent results regarding this invariant, along with various conjectures that have guided some of my own forays into this still-mysterious territory.

MC 5501