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Rigidity and almost rigidity of critical Sobolev embeddings on RCD spaces

Francesco Nobili
Tuesday, June 8, 2021 - 11:30 to 12:30
There has been a growing interest around Sobolev embeddings, especially to determine the values of the optimal constants and the shape of extremal functions. In this seminar, we focus on critical Sobolev embeddings on smooth compact Riemannian manifolds and establish a new rigidity result, which roughly states that an $n$-dimensional Riemannian manifold with Ricci curvature bounded below by $n-1$ and having the same optimal Sobolev constant of the $n$-sphere must be isometric to the $n$-sphere.
We carry out this analysis in the more general class of spaces satisfying the so-called ${\rm RCD}$-condition, namely metric measure spaces with synthetic Ricci lower bounds and dimension upper bounds, where we are also able to provide an almost rigidity version of the main rigidity result. Our arguments are based on a new Euclidean Polya-Szego inequality and on a variant of Lion's Concentration Compactness principle with varying ambient space. This is joint work with Ivan Yuri Violo.

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