| Résume | One of the most studied problems in Differential Geometry is the existence of Riemannian metrics with constant scalar curvature. In the 1980s, the celebrated solution of the Yamabe problem by Trudinger, Aubin, and Schoen established that on a closed manifold one can always find a constant scalar curvature metric, in each conformal class of Riemannian metrics. In the context of Kähler geometry, however, conformal methods can not be directly applied and the question of existence of constant scalar curvature Kähler metrics is still largely open. In this talk, I will present some progress in an ongoing project with Abdellah Lahdili (Université du Québec à Montréal) and Eveline Legendre (Université Lyon 1) in which we propose an approach to the existence of constant scalar curvature Kähler metrics that uses tools coming from the solution of the CR-Yamabe problem, most notably a version of the Einstein-Hilbert functional. I will explain how our methods can be used to prove that (a version of) the Yamabe Invariant detects a well-know algebraic obstruction to the existence of constant scalar curvature Kähler metrics, K-semistability. |
