In the 70s, Aubin computed a notion of optimal Sobolev constant on an compact Riemannian manifold. Its value is a central question in the celebrated AB program and turns out to be tightly linked to the problem of existence of solutions to the generalized Yamabe equation.

We review this program in the framework of possibly non-smooth spaces with synthetic Ricci lower bounds and establish existence of solutions for the generalized Yamabe equation and a new continuity result for the generalized Yamabe constant under measure Gromov-Hausdorff convergence. Our arguments are based on a novel Lions’ concentration compactness principle under varying spaces. Based on joint work with I. Y. Violo.