Letizia Branca: Existence results for the Bach tensor on 4D closed Riemannian manifolds

The Bach tensor is a fourth-order, symmetric 2-tensor; this geometric quantity was introduced by Bach in the context of conformal relativity and it has been widely studied on four-dimensional Riemannian manifolds, where it enjoys important conformal properties. In fact, it is closely related to the so-called Weyl functional, whose critical points are Riemannian metrics with vanishing Bach tensor, called Bach-flat metrics. However, very little is known about the relations between the Bach tensor and the topology of the underlying manifold. In this talk, we prove that on every closed 4D Riemannian manifold, there exists a metric whose Bach tensor is pinched by the scalar curvature (regardless of its topology), exploiting an Aubin’s deformation of the metric. Time permitting, we will also discuss the existence, in certain conformal classes, of critical points of a Bach functional on every closed 4D manifold, constructing unobstructed metrics generalizing the Bach-flat condition.
This talk is based on joit works with G. Catino and D. Dameno