In CD(N-1,N) spaces with near-sharp Sobolev constant, the pushforward measure of any almost-extremal is close in Wasserstein distance to an Aubin-Talenti bubble distribution, with sharp √ε rate.
Ambrosio , Calculus, heat flow and curvature-dimension bounds in metric measure spaces , in Proceedings of the I nternational C ongress of M athematicians--- R io de J aneiro 2018
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Quantitative stability in distribution for the Sobolev inequality under curvature dimension condition
In CD(N-1,N) spaces with near-sharp Sobolev constant, the pushforward measure of any almost-extremal is close in Wasserstein distance to an Aubin-Talenti bubble distribution, with sharp √ε rate.