Stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds

TL;DR

This paper investigates the stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature bounds, showing that near-extremal functions are close to extremals in both spherical and Euclidean settings, with applications to Yamabe metrics.

Contribution

It establishes quantitative stability results for Sobolev inequalities under Ricci curvature conditions, extending to singular spaces and applications to Yamabe problem.

Findings

Near-extremal functions are close to extremals on spheres.

Stability results hold for manifolds with non-negative Ricci curvature.

Applications include stability of Yamabe metrics.

Abstract

We study the qualitative stability of two classes of Sobolev inequalities on Riemannian manifolds. In the case of positive Ricci curvature, we prove that an almost extremal function for the sharp Sobolev inequality is close to an extremal function of the round sphere. In the setting of non-negative Ricci curvature and Euclidean volume growth, we show an analogous result in comparison with the extremal functions in the Euclidean Sobolev inequality. As an application, we deduce a stability result for minimizing Yamabe metrics. The arguments rely on a generalized Lions' concentration compactness on varying spaces and on rigidity results of Sobolev inequalities on singular spaces.