Triangles and triple products of Laplace eigenfunctions

TL;DR

This paper investigates the behavior of triple products of Laplace eigenfunctions on compact manifolds, revealing their concentration properties are linked to geometric triangle configurations, and improves existing bounds in this area.

Contribution

It establishes a connection between eigenfunction triple products and geometric triangle configurations, enhancing understanding of their concentration and providing improved bounds.

Findings

Triple products are concentrated according to triangle configurations.

A small proportion of triple products occur outside the triangle inequality regime.

The results refine previous bounds by Lu, Sogge, and Steinerberger.

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.