论文标题
将本征限制限制到完全测量的子曼群
Restriction of eigenfunctions to totally geodesic submanifolds
论文作者
论文摘要
本文涉及紧凑的Riemannian歧管$（M，G）$（M，G）$（M，G）$的两种类型的限制$ ϕ_j $：首先，我们限制在submanifold $ h \ subset m $中，并扩展限制$γ_hϕ_j $ eigenfunctions $ e_k $ e_k $ h $ h $ h $。然后，我们将傅立叶限制在$ h $的特征值的短时间间隔$γ_hϕ_j $。 $ m $的拉普拉斯特征值表示为$λ_j^2 $，$ h $的laplace值表示为$μ_k^2 $。 除非$ h $ - 特征值位于[-λ_j，λ_j] $中的间隔$μ_k\，否则傅立叶系数可以忽略不计。短窗口具有$ |μ_k -cλ_j| <ε$。目的是获得$γ_Hϕ_j $的傅立叶系数的渐近学和估计，并查看它们如何随$ c $而变化。在与E. L. Wyman和Y. XI的先前工作中，我们以$ 0 <c <1 $的价格获得了此类窗口中$（μ_k，λ_j）$的渐近（μ_k，λ_j）$。在本文中，当$ c = 1 $和$ h $完全是测量时，我们会获得“边缘”渐近造。数量级和领先系数与$ c <1 $的情况大不相同。特别是，它们取决于$ h $的尺寸。我们解释了如何弥合批量结果和边缘结果。
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.
