GTA Seminar | Speaker: Diego Mojón-Álvarez (University Santiago de Compostela, Spain)
 

Title: The weighted Einstein condition and its geometric implications

Abstract: Smooth metric measure spaces are obtained by modifying the usual volume form of a Riemannian manifold (M,g) through a smooth density function f. These structures, also referred to as manifolds with density, appear in a variety of settings, including functional inequalities and geometric flows, and lead to important geometric equations such as gradient Ricci solitons and quasi-Einstein metrics. In this talk, we will focus on a natural generalization of the latter, known as the weighted Einstein condition, and explore its impact on the geometry of the underlying manifold. To that end, we will go over some classification results under assumptions related to weighted conformal geometry. In particular, we will analyze a modified harmonicity condition for the weighted counterpart of the Weyl tensor, as well as the properties of weighted conformal classes that admit multiple nonhomothetic weighted Einstein structures. This is joint work with Miguel Brozos-Vázquez and Eduardo García-Río.