It seems that the $k^{th}$ largest eigenvalue of the intergral operator induced on $S^n$ by the Gaussian kernel, $e^{-\frac{\vert \vec{x} - \vec{y} \vert _2^2}{2\sigma^2}}$ decays as $k^{-k}$. This is an extremely fast decay as compared to what happens on say $\mathbb{R}^n$.
Are there other examples of compact manifolds (or any generic understanding) of where such fast or faster decays happen for the Gaussian (or any other!) kernel?
