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 \[ \langle e_λe_μ, e_ν\rangle = \int e_λe_μ\overline{e_ν} \, dV. \] 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.

29 pages, 1 figure