The problem of the stability of the Sobolev inequality in the Euclidean space asks how close is a function achieving almost equality,

to the class of minimizers. This question has received a lot of attention in recent years and is now well understood. On the other hand,

in the setting of Riemannian manifolds very few is known except in specific cases. In this talk, we show that almost extremal functions

for the sharp Sobolev inequality on a Riemannian manifolds with a positive lower bound on the Ricci curvature, are close to extremal

functions of the round sphere. A similar result for non-negative Ricci curvature and maximal volume growth is also presented. The

results rely on rigidity properties of the Sobolev inequality in RCD spaces and on a generalized version of Lions' concentration compactness in this setting.

This is joint work with Francesco Nobili.