Geodesic restrictions for the Casimir operator

On logarithmic improvements of critical geodesic restriction bounds in the presence of nonpositive curvature

We consider upper bounds on the growth of $L^p$ norms of restrictions of eigenfunctions and quasimodes to geodesic segments in a nonpositively curved manifold in the high frequency limit. This sharpens results of Chen and Sogge as well as Xi and Zhang, which showed that the crux of the problem is to establish bounds on the mixed partials of the distance function on the covering manifold restricted to geodesic segments. The innovation in this work is the development of a formula for the third variation of arc length on the covering manifold, which allows for a coordinate free expressions of the…

The $L^2$ restriction of the Eisenstein series to a geodesic segment

We study the $L^2$ norm of the Eisenstein series $E(z,1/2+iT)$ restricted to a segment of a geodesic connecting infinity and an arbitrary real. We conjecture that on slightly thickened geodesics of this form, the Eisenstein series satifies restricted QUE. We prove a lower bound that matches this predicted asymptotic. We also prove an upper bound that nearly matches the lower bound assuming the Riemann Hypothesis (unconditionally, the sharp upper bound holds for almost all $T$). Finally, we show the restricted QUE conjecture for geodesics with rational endpoints.

Restriction of eigenfunctions to totally geodesic submanifolds

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 e…

Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions $ϕ_j$ 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, $$ N^{c} _{ε, H }(λ): = \sum_{j, λ_j \leq λ} \sum_{k: |μ_k - c λ_j | < ε} \left| \int_{H} ϕ_j \overline{ψ_k}dV_H \right|^2 $$ where $\{λ_j\}$ are the eigenvalues of $\sqrt{-Δ}_M,$ and $\{(μ_k, ψ_k)\}$ are the eigenvalues, resp. eigenfunctions, of $\sqrt{-Δ}_H$. The inner sums represent the `jumps' of $ N^{c} _{ε, H }(λ)$ and reflect the geometry o…

Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds

We give estimates for the $L^p$ norm ($2\leq p \leq +\infty$) of the restriction to a curve of the eigenfunctions of the Laplace Beltrami operator on a Riemannian surface. If the curve is a geodesic, we show that on the sphere these estimates are sharp. If the curve has non vanishing geodesic curvature, we can improve our results. We also show how our approach apply to higher dimensional manifolds.

Maass形式在超曲面和测地管上的限制

Restrictions of Maass forms on $\mathrm{SL}(2,\mathbb{C})$ to hyperbolic surfaces and geodesic tubes

本文通过算术放大方法改进了Maass形式在超曲面和测地管上的L²界，获得了更优的功率节省结果。

Improved critical eigenfunction restriction estimates on Riemannian manifolds with constant negative curvature

We show that one can obtain logarithmic improvements of $L^2$ geodesic restriction estimates for eigenfunctions on 3-dimensional compact Riemannian manifolds with constant negative curvature. We obtain a $(\logλ)^{-\frac12}$ gain for the $L^2$-restriction bounds, which improves the corresponding bounds of Burq, Gérard and Tzvetkov, Hu, Chen and Sogge. We achieve this by adapting the approaches developed by Chen and Sogge, Blair and Sogge, Xi and the author. We derive an explicit formula for the wave kernel on 3D hyperbolic space, which improves the kernel estimates from the Hadamard parametrix…

Improved bounds for restricted projection families via weighted Fourier restriction

It is shown that if $A \subseteq \mathbb{R}^3$ is a Borel set of Hausdorff dimension $\dim A \in (3/2,5/2)$, then for a.e. $θ\in [0,2π)$ the projection $π_θ(A)$ of $A$ onto the 2-dimensional plane orthogonal to $\frac{1}{\sqrt{2}}(\cos θ, \sin θ, 1)$ satisfies $\dim π_θ(A) \geq \max\left\{\frac{4\dim A}{9} + \frac{5}{6},\frac{2\dim A+1}{3} \right\}$. This improves the bound of Oberlin and Oberlin, and of Orponen and Venieri, for $\dim A \in (3/2,5/2)$. More generally, a weaker lower bound is given for families of planes in $\mathbb{R}^3$ parametrised by curves in $S^2$ with nonvanishing geodes…

曲面上 Neumann 数据的 $L^p$ 限制界

The $L^p$ restriction bounds for Neumann data on surface

本文研究了紧致二维光滑 Riemannian 流形上 Laplacian 特征函数沿测地线的 Neumann 数据的 $L^p$ 限制界，通过 $T$-$T^*$ 论证和 Van De Corput 定理证明了最优界 $O(λ^{-\frac{1}p+\frac{3}2})$。

Non-concentration and restriction bounds for Neumann eigenfunctions of piecewise $C^{\infty}$ bounded planar domains

Let $(Ω,g)$ be a piecewise-smooth, bounded convex domain in $\R^2$ and consider $L^2$-normalized Neumann eigenfunctions $ϕ_λ$ with eigenvalue $λ^2$ and $u_λ:= ϕ_λ |_{\partial Ω}$ the associated Dirichlet data (ie. boundary restriction of $ϕ_λ$). Our first main result (Theorem \ref{T:non-con}) is a small-scale {\em non-concentration} estimate: We prove that for {\em any} $x_0 \in \overlineΩ,$ (including boundary corner points) and any $δ\in [0,1),$ $$ \| ϕ_h \|_{B(x_0,λ^{-δ})\cap Ω} = O(λ^{-δ/2}).$$ Our subsequent results involve applications of the nonconcentration estimate to upper bounds for…

On eigenfunction restriction estimates and $L^4$-bounds for compact surfaces with nonpositive curvature

Let $(M,g)$ be a two-dimensional compact boundaryless Riemannian manifold with nonpostive curvature, then we shall give improved estimates for the $L^2$-norms of the restrictions of eigenfunctions to unit-length geodesics, compared to the general results of Burq, Gérard and Tzvetkov \cite{burq}. By earlier results of Bourgain \cite{bourgainef} and the first author \cite{Sokakeya}, they are equivalent to improvements of the general $L^p$-estimates in \cite{soggeest} for $n=2$ and $2<p<6$. The proof uses the fact that the exponential map from any point in $x_0\in M$ is a universal covering map f…

正交系统的尖锐谱簇限制界

Sharp spectral-cluster restriction bounds for orthonormal systems

本文针对紧致黎曼流形上单位宽度谱簇内的正交归一特征函数系，建立了其在光滑子流形上的尖锐谱簇限制界，发现余维数1时存在新的维数无关临界点，还改进了曲线情形的估计。