Triangles and triple products of Laplace eigenfunctions

Abstract

Consider an $L^2$-normalized Laplace-Beltrami eigenfunction $e_\lambda$ on a compact, boundary-less Riemannian manifold with $\Delta e_\lambda = -\lambda^2 e_\lambda$. We study eigenfunction triple products \[ \langle e_\lambda e_\mu, e_\nu \rangle = \int e_\lambda e_\mu \overline{e_\nu} \, dV. \] We show the overall $\ell^2$-concentration of these triple products is determined by the measure of some set of configurations of triangles with side lengths equal to the frequencies $\lambda,\mu,$ and $\nu$. A rapidly vanishing proportion of this mass lies in the `classically forbidden' regime where $\lambda, \mu,$ and $\nu$ fail to satisfy the triangle inequality. As a consequence, we improve a result by Lu, Sogge, and Steinerberger.