Rigidity and almost rigidity of Sobolev inequalities on compact spaces with lower Ricci curvature bounds

Abstract

AbstractWe prove that ifMis a closedn-dimensional Riemannian manifold,$$n \ge 3$$n≥3, with$$\mathrm{Ric}\ge n-1$$Ric≥n-1and for which the optimal constant in the critical Sobolev inequality equals the one of then-dimensional sphere$$\mathbb {S}^n$$Sn, thenMis isometric to$$\mathbb {S}^n$$Sn. 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 the$$\mathrm {RCD}$$RCD-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 compact$$\mathrm {CD}$$CDspace, 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 of$$\mathrm {RCD}$$RCDspaces and on a Pólya–Szegő inequality of Euclidean-type in$$\mathrm {CD}$$CDspaces. 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 the$$\mathrm {RCD}$$RCD-setting.