Restriction of eigenfunctions to totally geodesic submanifolds

Abstract

This article is about two types of restrictions of eigenfunctions $ϕ_j$ on a compact Riemannian manifold $(M,g)$: First, we restrict to a submanifold $H \subset M$, and expand the restriction $γ_H ϕ_j$ in eigenfunctions $e_k$ of $H$. We then Fourier restrict $γ_H ϕ_j$ to a short interval of eigenvalues of $H$. Laplace eigenvalues of $M$ are denoted $λ_j^2$ and those of $H$ are denoted $μ_k^2$. The Fourier coefficients are negligible unless the $H$- eigenvalues lie in the interval $μ_k \in [-λ_j, λ_j]$. The short windows have the form $|μ_k - c λ_j| < ε$. The goal is to obtain asymptotics and estimates of the Fourier coefficients of $γ_H ϕ_j$ and to see how they vary with $c$. In prior work with E. L. Wyman and Y. Xi, we obtained asymptotics for sums over $(μ_k, λ_j)$ in such windows for $0 < c < 1$. In this article, we obtain `edge' asymptotics when $c=1$ and $H$ is totally geodesic. The order of magnitude and leading coefficient are very different from the case $c<1$. In particular, they depend on the dimension of $H$. We explain how to bridge the bulk results and edge results.