@booklet {2021, title = {Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds}, year = {2021}, abstract = {

We prove that ifMis a closedn-dimensional Riemannian manifold,n>=3, withRic>=n-1and for which the optimal constant in the critical Sobolev inequality equals the one of then-dimensional sphereSn, thenMis isometric toSn. An almost-rigidity result is also established, saying that if equality is almost achieved, thenMis close in the measure Gromov-Hausdorff sense to a spherical suspension. These statements are obtained in theRCD-setting of (possibly non-smooth) metric measure spaces satisfying synthetic lower Ricci curvature bounds.
An independent result of our analysis is the characterization of the best constant in the Sobolev inequality on any compactCDspace, extending to the non-smooth setting a classical result by Aubin. Our arguments are based on a new concentration compactness result for mGH-converging sequences ofRCDspaces and on a Polya-Szego inequality of Euclidean-type inCDspaces.
As an application of the technical tools developed we prove both an existence result for the Yamabe equation and the continuity of the generalized Yamabe constant under measure Gromov-Hausdorff convergence, in theRCD-setting.
}, author = {Francesco Nobili and Ivan Yuri Violo} }