Let $(M, g)$ be a compact Riemannian manifold, and let $\lambda^2$, $\varphi_\lambda$ represent eigenvalues and eigenfunctions respectively of the Laplacian $\Delta$, that is, $-\Delta \varphi_\lambda = \lambda^2\varphi_\lambda$. It is known that the Gaussian beams like highest weight spherical harmonics decay exponentially away from a stable elliptic orbit. It is also known that in general, $$ \sup_{B(p, h)} \varphi_\lambda \leq C_1e^{-C_2\lambda},$$ where $p \in M, h = \frac{1}{\lambda}$, and $C_1, C_2$ are constants independent of $\lambda$.
My question is, are there other examples known (apart from highest weight spherical harmonics) where the exponential bound above is saturated? In the negative curvature (ergodic) case, do better bounds hold in general? This is mainly a reference request.
