Welcome to the website of the seminar organized by the Differential Geometry groups of Università di Torino and Politecnico di Torino.
For any information, please send an e-mail to dgseminar.torino@gmail.com.
The talks will take place either at the Deparment of Mathematics "G. Peano" (Università di Torino) or at the Department of Mathematical Sciences "G.L. Lagrange" (Politecnico di Torino).
NEXT TALK
25/11/2025, 2.30 AM, Sala S, Dipartimento di Matematica "G. Peano", Università di Torino
Abstract: The Bach tensor is a fourth-order, symmetric 2-tensor which has been extensively studied in the context of Riemannian four-manifolds, where it happens to enjoy important conformal properties: indeed, this curvature quantity naturally appears in the framework of Conformal Relativity and it is intimately related to the so-called Weyl functional, whose critical points are exactly the Riemannian metrics with vanishing Bach tensor, called Bach-flat metrics. A long-standing open conjecture in Riemannian Geometry states that, on every (closed) four-manifold, there exists a Bach-flat metric: in general, 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 four-manifold, regardless of its topology, there exists a Riemannian metric with “small” Bach tensor, i.e. a metric whose Bach tensor is uniformly controlled by the scalar curvature. Our approach, inspired by the work of Gursky, is based on the resolution of a modified Yamabe problem and on local deformations, first introduced by Aubin, of reference metrics constructed in a previous joint work with Catino and Mastrolia. Time permitting, we will also sketch the proof of the existence of critical points, in many given conformal classes, of a Bach functional on every closed four-manifold, henceforth constructing unobstructed metrics which generalize the Bach-flat condition.
This is based on joint works with L. Branca and G. Catino.
FORTHCOMING TALKS
TBA