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

Abstract We prove that if M is a closed n -dimensional Riemannian manifold, $$n \ge 3$$ n≥3 , with $$\mathrm{Ric}\ge n-1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Ric≥n-1 and for which the optimal constant in critical Sobolev inequality equals one of sphere $$\mathbb {S}^n$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">Sn then isometric to . An almost-rigidity result also established, saying equality almost achieved, close measure Gromov–Hausdorff sense spherical suspension. These statements are obtained $$\mathrm {RCD}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">RCD -setting (possibly non-smooth) metric spaces satisfying synthetic lower Ricci curvature bounds. independent our analysis characterization best on any compact {CD}$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">CD space, extending non-smooth setting classical by Aubin. Our arguments based new concentration compactness mGH-converging sequences Pólya–Szegő Euclidean-type spaces. As an application technical tools developed we both existence Yamabe equation continuity generalized under convergence, -setting.