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

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arxiv 2412.05935 v2 pith:QLV4IA7I submitted 2024-12-08 math.AP math.DGmath.MG

Francesco Nobili This is my paper
classification math.APmath.DGmath.MG
keywords stabilitymanifoldsriccisobolevboundscurvatureinequalitieslower
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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.

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