Restriction of eigenfunctions to totally geodesic submanifolds

TL;DR

This paper investigates how eigenfunctions on a compact Riemannian manifold behave when restricted to totally geodesic submanifolds, especially at the 'edge' case where the eigenvalue ratio c equals 1, revealing new asymptotic behaviors.

Contribution

It provides the first analysis of eigenfunction restrictions at the edge case c=1 for totally geodesic submanifolds, highlighting dimension-dependent asymptotics.

Findings

Edge asymptotics differ significantly from bulk cases.

Leading coefficients depend on the dimension of the submanifold.

Bridges between bulk and edge asymptotic regimes are established.

Abstract

This article is about two types of restrictions of eigenfunctions $\phi_j$ on a compact Riemannian manifold $(M,g)$: First, we restrict to a submanifold $H \subset M$, and expand the restriction $\gamma_H \phi_j$ in eigenfunctions $e_k$ of $H$. We then Fourier restrict $\gamma_H \phi_j$ to a short interval of eigenvalues of $H$. Laplace eigenvalues of $M$ are denoted $\lambda_j^2$ and those of $H$ are denoted $\mu_k^2$. The Fourier coefficients are negligible unless the $H$- eigenvalues lie in the interval $\mu_k \in [-\lambda_j, \lambda_j]$. The short windows have the form $|\mu_k - c \lambda_j| < \epsilon$. The goal is to obtain asymptotics and estimates of the Fourier coefficients of $\gamma_H \phi_j$ and to see how they vary with $c$. In prior work with E. L. Wyman and Y. Xi, we obtained asymptotics for sums over $(\mu_k, \lambda_j)$ 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.