Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

TL;DR

This paper investigates the asymptotic behavior of Fourier coefficients of Laplace eigenfunction restrictions to submanifolds, revealing geometric structures called c-bi-angles that influence the eigenfunction interactions.

Contribution

It introduces the concept of c-bi-angles and analyzes their impact on the asymptotics of Fourier coefficient sums, refining previous results by incorporating geodesic bi-angle geometry.

Findings

Identification of c-bi-angles affecting Fourier coefficient jumps

Sharp estimates on the size and variation of jumps in sums

Illustrations using subspheres of spheres and subtori of tori

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, $$ N^{c} _{\epsilon, H }(\lambda): = \sum_{j, \lambda_j \leq \lambda} \sum_{k: |\mu_k - c \lambda_j | < \epsilon} \left| \int_{H} \phi_j \overline{\psi_k}dV_H \right|^2 $$ where $\{\lambda_j\}$ are the eigenvalues of $\sqrt{-\Delta}_M,$ and $\{(\mu_k, \psi_k)\}$ are the eigenvalues, resp. eigenfunctions, of $\sqrt{-\Delta}_H$. The inner sums represent the `jumps' of $ N^{c} _{\epsilon, H }(\lambda)$ and reflect the geometry of geodesic c-bi-angles with one leg on $H$ and a second leg on $M$ with the same endpoints and compatible initial tangent vectors $\xi \in S^c_H M, \pi_H \xi \in B^* H$, where $\pi_H \xi$ is the orthogonal projection of $\xi$ to $H$. A c-bi-angle occurs when $\frac{|\pi_H \xi|}{|\xi|} = c$. Smoothed sums in $\mu_k$ are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as $\epsilon$ varies, at certain values of $\epsilon$ related to periodicities in the c-bi-angle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of the previous article (arXiv:2011.11571) where the inner sums run over $k: | \frac{\mu_k}{\lambda_j} - c| \leq \epsilon$ and where geodesic bi-angles do not play a role.