GEODESIC BIANGLES AND FOURIER COEFFICIENTS OF RESTRICTIONS OF EIGENFUNCTIONS

Abstract

This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions ${\phi }_j$ of a compact Riemannian manifold to a submanifold $H\subset M$. We fix a number $c\in (0,1)$ and study the asymptotics of the thin sums,

where $\{{\lambda }_j\}$ are the eigenvalues of ${\sqrt{-\mathrm{\Delta }}}_M$, and $\{({\mu }_k,{\psi }_k)\}$ are the eigenvalues and the corresponding eigenfunctions of ${\sqrt{-\mathrm{\Delta }}}_H$. The inner sums represent the “jumps” of $N_{𝜖,H}^c(\lambda )$ and reflect the geometry of geodesic $c$-biangles with one leg on $H$ and a second leg on $M$ with the same endpoints and compatible initial tangent vectors $\xi \in S_H^{c}M$, ${\pi }_H\xi \in B^{\ast }H$, where ${\pi }_H\xi$ is the orthogonal projection of $\xi$ to $H$. A $c$-biangle occurs when $\left|{\pi }_H\xi \right|∕\left|\xi \right|=c$. Smoothed sums in ${\mu }_k$ are also studied and give sharp estimates on the jumps. The jumps themselves may jump as $𝜖$ varies, at certain values of $𝜖$ related to periodicities in the $c$-biangle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of our previous article, where the inner sums run over $k$ such that $|{\mu }_k∕{\lambda }_j-c|\le 𝜖$ and where geodesic biangles do not play a role.