On the stability of Sobolev inequalities on Riemannian manifolds

Speakers: Francesco Nobili\n\nAbstract. The stability problem for the sharp Sobolev inequality in Euclidean space asks whether functions that nearly achieve equality must be close to the family of optimisers. This question has attracted significant attention in recent years and is now well understood. In contrast, on Riemannian manifolds the presence of curvature introduces new phenomena leading to challenging analysis.\nIn this seminar, we present recent results on the stability of Sobolev inequalities on Riemannian manifolds, obtained in collaboration with D. Parise and I. Y. Violo. After a brief overview of the Euclidean theory, we discuss both quantitative and qualitative stability results for several Sobolev-type inequalities, of geometric and functional nature. Special attention will be devoted to the role of lower Ricci curvature bounds, highlighting the use of nonsmooth techniques such as generalized concentration compactness principles.\n\nhttps://indico.math.cnrs.fr/event/16066/