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

Abstract

Abstract We prove that if M is a closed n -dimensional Riemannian manifold, $$n \ge 3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>n</mml:mi><mml:mo>≥</mml:mo><mml:mn>3</mml:mn></mml:mrow></mml:math> , with $$\mathrm{Ric}\ge n-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>Ric</mml:mi><mml:mo>≥</mml:mo><mml:mi>n</mml:mi><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> and for which the optimal constant in the critical Sobolev inequality equals the one of the n -dimensional sphere $$\mathbb {S}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> , then M is isometric to $$\mathbb {S}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mi>n</mml:mi></mml:msup></mml:math> . An almost-rigidity result is also established, saying that if equality is almost achieved, then M is close in the measure Gromov–Hausdorff sense to a spherical suspension. These statements are obtained in the $$\mathrm {RCD}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>RCD</mml:mi></mml:math> -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 $$\mathrm {CD}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>CD</mml:mi></mml:math> space, 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 $$\mathrm {RCD}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>RCD</mml:mi></mml:math> spaces and on a Pólya–Szegő inequality of Euclidean-type in $$\mathrm {CD}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>CD</mml:mi></mml:math> spaces. 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 $$\mathrm {RCD}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>RCD</mml:mi></mml:math> -setting.

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