An overview of the stability of Sobolev inequalities on Riemannian manifolds with Ricci lower bounds

TL;DR

This paper reviews recent advances in understanding the stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature bounds, highlighting techniques, specific results, and open problems in the field.

Contribution

It provides a comprehensive overview of methods and results on Sobolev inequality stability, including a detailed proof for manifolds with non-negative Ricci curvature and Euclidean volume growth.

Findings

Stability of Sobolev inequalities on certain Riemannian manifolds established.

Effective combination of smooth and non-smooth geometric techniques.

Open problems and future directions discussed.

Abstract

We review recent results regarding the problem of the stability of Sobolev inequalities on Riemannian manifolds with Ricci curvature lower bounds. We shall describe techniques and methods from smooth and non-smooth geometry, the fruitful combination of which revealed particularly effective. Furthermore, we present a self-contained overview of the proof of the stability of the Sobolev inequality on manifolds with non-negative Ricci curvature and Euclidean volume growth, adopting a direct strategy tailored to this setting. Finally, we discuss related stability results and present some open problems.