Quantitative stability in distribution for the Sobolev inequality under curvature dimension condition

The goal of this note is to investigate quantitative stability properties of the critical Sobolev inequality in $\mathsf{C}\mathsf{D}(N-1,N)$ metric measure spaces. Assuming that the optimal constant for the inequality is almost the same as the one of the round sphere, we show that the cumulative distribution of any almost extremal function is close, in Wasserstein distance, to the one of an Aubin-Talenti bubble on the round sphere. We obtain similar results for the log Sobolev inequality and the spectral gap under various curvature and dimension assumptions. In all cases we obtain a quantitative stability with sharp exponent.