Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{ϕ_λ\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-Δ_gϕ_λ=λ^2 ϕ_λ$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions on the pair $(M,H)$, even when $H=\{x\}$, for which $$ \Big|\int_Hϕ_λdσ_H\Big|=O\Big(\frac{λ^{\frac{k-1}{2}}}{\sqrt{\log λ}}\Big)\qquad \text{or}\qquad |ϕ_λ(x)|=O\Big(\frac{λ^{\frac{n-1}{2}}}{\sqrt{\log λ}}\Big), $$ as $λ\to \infty$. These conditions require no global assumption on the manifold $M$ and instead relate to the structure of the set of recurrent directions in the unit normal bundle to $H$. Our results extend all previously known conditions guaranteeing improvements on averages, including those on sup-norms. For example, we show that if $(M,g)$ is a surface with Anosov geodesic flow, then there are logarithmically improved averages for any $H\subset M$. We also find weaker conditions than having no conjugate points which guarantee $\sqrt{\log λ}$ improvements for the $L^\infty$ norm of eigenfunctions. Our results are obtained using geodesic beam techniques, which yield a mechanism for obtaining general quantitative improvements for averages and sup-norms.
For $2\leq p<4$, we study the $L^p$ norms of restrictions of eigenfunctions of the Laplace-Beltrami operator on smooth compact $2$-dimensional Riemannian manifolds. Burq, Gérard, and Tzvetkov \cite{BurqGerardTzvetkov2007restrictions}, and Hu \cite{Hu2009lp} found the eigenfunction estimates restricted to a curve with nonvanishing geodesic curvatures. We will explain how the proof of the known estimates helps us to consider the case where the given smooth compact Riemannian manifold has nonpositive sectional curvatures. For $p=4$, we will also obtain a logarithmic analogous estimate, by using a…
If $(M,g)$ is a compact Riemannian surface then the integrals of $L^2(M)$-normalized eigenfunctions $e_j$ over geodesic segments of fixed length are uniformly bounded. Also, if $(M,g)$ has negative curvature and $γ(t)$ is a geodesic parameterized by arc length, the measures $e_j(γ(t))\, dt$ on $\R$ tend to zero in the sense of distributions as the eigenvalue $\la_j\to \infty$, and so integrals of eigenfunctions over periodic geodesics tend to zero as $\la_j\to \infty$. The assumption of negative curvature is necessary for the latter result.
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.
We consider restrictions along closed geodesics and geodesic circles for eigenfunctions of the Laplace-Beltrami operator on a compact hyperbolic Riemann surface. We obtain a non-trivial bound on the L^2-norm of such restrictions as the eigenvalue tends to infinity. We use methods from the theory of automorphic functions and in particular the uniqueness of invariant functionals on irreducible unitary representations of PGL(2,R).
Let X be an arithmetic hyperbolic surface, ψa Hecke-Maass form, and l a geodesic segment on X. We obtain a power saving over the local bound of Burq-Gérard-Tzvetkov for the L^2 norm of ψrestricted to l, by extending the technique of arithmetic amplification developed by Iwaniec and Sarnak. We also improve the local bounds for various Fourier coefficients of ψalong l.
We estimate the $L^2$ norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace-Beltrami operator on the standard flat torus $\mathbb{T}^d$, $d\ge2$. We reduce getting correct bounds to counting lattice points in the intersection of some $ν$-transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In particular, we verify the conjecture of Bourgain-Rudnick on $L^2$-restriction estimates for rational hyperplanes. On $\mathbb{T}^2$, we prove the uniform $L^2$ restriction bounds …
We use the Gauss-Bonnet theorem and the triangle comparison theorems of Rauch and Toponogov to show that on compact Riemann surfaces of negative curvature period integrals of eigenfunctions $e_λ$ over geodesics go to zero at the rate of $O((\logλ)^{-1/2})$ if $λ$ are their frequencies. As discussed in \cite{CSPer}, no such result is possible in the constant curvature case if the curvature is $\ge0$. Notwithstanding, we also show that these bounds for period integrals are valid provided that integrals of the curvature over all geodesic balls of radius $r\le 1$ are pinched from above by $-δr^N$ …
This paper concerns the concentration of Dirichlet eigenfunctions of the Laplacian on a compact two-dimensional Riemannian manifold with strictly geodesically concave boundary. We link three inequalities which bound the concentration in different ways. We also prove one of these inequalities, which bounds the L^p norms of the restrictions of eigenfunctions to broken geodesics.
In this article we develop new techniques for studying concentration of Laplace eigenfunctions $ϕ_λ$ as their frequency, $λ$, grows. The method consists of controlling $ϕ_λ(x)$ by decomposing $ϕ_λ$ into a superposition of geodesic beams that run through the point $x$. Each beam is localized in phase-space on a tube centered around a geodesic whose radius shrinks slightly slower than $λ^{-\frac{1}{2}}$. We control $ϕ_λ(x)$ by the $L^2$-mass of $ϕ_λ$ on each geodesic tube and derive a purely dynamical statement through which $ϕ_λ(x)$ can be studied. In particular, we obtain estimates on $ϕ_λ(x)$…
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…
We prove a couple of new endpoint geodesic restriction estimates for eigenfunctions. In the case of general 3-dimensional compact manifolds, after a $TT^*$ argument, simply by using the $L^2$-boundedness of the Hilbert transform on $\R$, we are able to improve the corresponding $L^2$-restriction bounds of Burq, Gérard and Tzvetkov and Hu. Also, in the case of 2-dimensional compact manifolds with nonpositive curvature, we obtain improved $L^4$-estimates for restrictions to geodesics, which, by Hölder's inequality and interpolation, implies improved $L^p$-bounds for all exponents $p\ge 2$. We do…
Let $(M,g)$ be a compact Riemannian surface with nonpositive sectional curvature and let $γ$ be a closed geodesic in $M$. And let $e_λ$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator $Δ_g$ with $-Δ_g e_λ= λ^2 e_λ$. Sogge, Xi, and Zhang showed using the Gauss-Bonnet theorem that $$ \int_γe_λ\, ds = O((\logλ)^{-1/2}),$$ an improvement over the general $O(1)$ bound. We show this integral enjoys the same decay for a wide variety of curves, where $M$ has nonpositive sectional curvature. These are the curves $γ$ whose geodesic curvature avoids, pointwise, the geodesic curvatur…
Consider $M$, a bounded domain in ${\mathbb R}^d$, which is a Riemanian manifold with piecewise smooth boundary and suppose that the billiard associated to the geodesic flow reflecting on the boundary acording to the laws of geometric optics is ergodic. We prove that the boundary value of the eigenfunctions of the Laplace operator with reasonable boundary conditions are asymptotically equidistributed in the boundary, extending previous results by Gérard, Leichtnam \cite{GeLe93-1} and Hassel, Zelditch \cite{HaZe02} obtained under the additional assumption of the convexity of $M$.
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…
Burq-Gérard-Tzvetkov and Hu established $L^p$ estimates ($2\le p\le \infty$) for the restriction of eigenfunctions to submanifolds. The estimates are sharp, except for the log loss at the endpoint $L^2$ estimates for submanifolds of codimension 2. It has long been believed that the log loss at the endpoint can be removed in general, while the problem is still open. So this paper is devoted to the study of sharp endpoint restriction estimates for eigenfunctions in this case. Chen and Sogge removed the log loss for the geodesics on 3-dimensional manifolds. In this paper, we generalize their resu…
Let $(M,g)$ be a smooth, compact Riemannian manifold and $\{ϕ_h\}$ an $L^2$-normalized sequence of Laplace eigenfunctions, $-h^2Δ_gϕ_h=ϕ_h$. Given a smooth submanifold $H \subset M$ of codimension $k\geq 1$, we find conditions on the pair $(\{ϕ_h\},H)$ for which $$ \Big|\int_Hϕ_hdσ_H\Big|=o(h^{\frac{1-k}{2}}),\qquad h\to 0^+. $$ One such condition is that the set of conormal directions to $H$ that are recurrent has measure $0$. In particular, we show that the upper bound holds for any $H$ if $(M,g)$ is surface with Anosov geodesic flow or a manifold of constant negative curvature. The results …
Let $(S^2,g)$ be a convex surface of revolution and $H \subset S^2$ the unique rotationally invariant geodesic. Let $φ^\ell_m$ be the orthonormal basis of joint eigenfunctions of $Δ_g$ and $\partial_θ$, the generator of the rotation action. The main result is an explicit formula for the weak-* limit of the normalized empirical measures, $Σ_{m = -\ell}^\ell ||φ^\ell_m||^2_{L^2(H)} δ_{\frac{m}{\ell}}(c)$ on $[-1,1]$. The explicit formula shows that, asymptotically, the $L^2$ norms of restricted eigenfunctions are minimal for the zonal eigenfunction $m = 0$, maximal for Gaussian beams $m = \pm 1$…
Let $(M,g)$ be a compact, 2-dimensional Riemannian manifold with nonpositive sectional curvature. Let $Δ_g$ be the Laplace-Beltrami operator corresponding to the metric $g$ on $M$, and let $e_λ$ be $L^2$-normalized eigenfunctions of $Δ_g$ with eigenvalue $λ$, i.e. \[ -Δ_g e_λ= λ^2 e_λ. \] We prove \[ \left| \int_{\mathbb R} b(t) e_λ(γ(t)) \, dt \right| = o(1) \quad \text{ as } λ\to \infty \] where $b$ is a smooth, compactly supported function on $\mathbb R$ and $γ$ is a curve parametrized by arc-length whose geodesic curvature $κ(γ(t))$ avoids two critical curvatures $\mathbf k(γ'^\perp(t))$ a…
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…
We work on the boundary $\partial M_τ$ of a Grauert tube of a closed, real analytic Riemannian manifold $M$. The Toeplitz operator $Π_τD_{\sqrtρ} Π_τ$ associated to the Reeb vector field is a positive, self-adjoint, elliptic operator on $H^2(\partial M_τ)$. We compute $λ\to \infty$ asymptotics under parabolic rescaling in a neighborhood of the geodesic (Reeb) flow $G^{t}_τ = \exp tΞ_{\sqrtρ}$ for the spectral projection kernel $Π_{χ, λ}$ associated to $Π_τD_{\sqrtρ} Π_τ$. We also compute scaling asymptotics for tempered sums of Husimi distributions (analytic continuations) on $\partial M_τ$ of…
We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold $(M,g)$ of dimension $n$, let $Π_λ$ denote the kernel of the spectral projector for the Laplacian, $\mathbb{1}_{[0,λ^2]}(-Δ_g)$. Assuming only that the set of near periodic geodesics over $W\subset M$ has small measure, we prove that as $λ\to \infty$ $$ \int_{W} Π_λ(x,x)dx=(2π)^{-n}\text{vol}_{\mathbb{R}^n}(B)\text{vol}_g(W)\,λ^n+O\Big(\frac{λ^{n-1}}{\log λ}\Big),$$ where $B$ is the unit ball. One consequence of this result is that the improved r…
We provide a necessary and sufficient condition that $L^p$-norms, $2<p<6$, of eigenfunctions of the square root of minus the Laplacian on 2-dimensional compact boundaryless Riemannian manifolds $M$ are small compared to a natural power of the eigenvalue $λ$. The condition that ensures this is that their $L^2$ norms over $O(λ^{-1/2})$ neighborhoods of arbitrary unit geodesics are small when $λ$ is large (which is not the case for the highest weight spherical harmonics on $S^2$ for instance). The proof exploits Gauss' lemma and the fact that the bilinear oscillatory integrals in Hörmander's proo…
In this short note we show that the lower bounds of Mangoubi on the inner radius of nodal domains can be improved for quantum ergodic sequences of eigenfunctions, according to a certain power of the radius of shrinking balls on which the eigenfunctions equidistribute. We prove such improvements using a quick application of our recent results (arXiv:1606.02057), which give modified growth estimates for eigenfunctions that equidistribute on small balls. Since by the results of Han and Hezari-Rivière small scale QE holds for negatively curved manifolds on logarithmically shrinking balls, we get l…
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…
We prove a Kakeya-Nikodym bound on eigenfunctions and quasimodes, which sharpens a result of the authors and extends it to higher dimensions. As in the prior work, the key intermediate step is to prove a microlocal version of these estimates, which involves a phase space decomposition of these modes which is essentially invariant under the bicharacteristic/geodesic flow. In a companion paper, it will be seen that these sharpened estimates yield improved $L^q(M)$ bounds on eigenfunctions in the presence of nonpositive curvature when $2 < q < \frac{2(d+1)}{d-1}$.
We prove an analogue of Sogge's local $L^p$ estimates for $L^p$ norms of restrictions of eigenfunctions to submanifolds, and use it to show that for quantum ergodic eigenfunctions one can get improvements of the results of Burq-Gérard-Tzvetkov, Hu, and Chen-Sogge. The improvements are logarithmic on negatively curved manifolds (without boundary) and by $o(1)$ for manifolds (with or without boundary) with ergodic geodesic flows. In the case of ergodic billiards with piecewise smooth boundary, we get $o(1)$ improvements on $L^\infty$ estimates of Cauchy data away from a shrinking neighborhood of…
Let $φ_λ$ be an eigenfunction of the Laplace-Beltrami operator on a smooth compact Riemannian manifold $(M,g)$, i.e., $Δ_g φ_λ + λφ_λ=0$. We show that $φ_λ$ satisfies a local Bernstein inequality, namely for any geodesic ball $B_g(x,r)$ in $M$ there holds: $\sup_{B_g(x,r)}|\nablaφ_λ|\leq C_δ\max\left\{\frac{\sqrtλ\log^{2+δ}λ}{r},λ\log^{2+δ}λ\right\}\sup_{B_g(x,r)}|φ_λ|$. We also prove analogous inequalities for solutions of elliptic PDEs in terms of the frequency function.
If $(M,g)$ is a compact real analytic Riemannian manifold, we give a necessary and sufficient condition for there to be a sequence of quasimodes of order $o(λ)$ saturating sup-norm estimates. In particular, it gives optimal conditions for existence of eigenfunctions satisfying maximal sup norm bounds. The condition is that there exists a self-focal point $x_0\in M$ for the geodesic flow at which the associated Perron-Frobenius operator $U_{x_0}: L^2(S_{x_0}^*M) \to L^2(S_{x_0}^*M)$ has a nontrivial invariant $L^2$ function. The proof is based on an explict Duistermaat-Guillemin-Safarov pre-tra…
Let $(M,g)$ be a compact $n$-dimensional Riemannian manifold without boundary and $e_λ$ be an $L^2$-normalized eigenfunction of the Laplace-Beltrami operator with respect to the metric $g$, i.e \[ -Δ_g e_λ= λ^2 e_λ\qquad \text{ and } \qquad \| e_λ\|_{L^2(M)} = 1. \] Let $Σ$ be a $d$-dimensional submanifold and $dμ$ a smooth, compactly supported measure on $Σ$. It is well-known (e.g. proved by Zelditch in far greater generality) that \[ \int_Σe_λ\, dμ= O(λ^\frac{n-d-1}{2}). \] We show this bound improves to $o(λ^\frac{n-d-1}{2})$ provided the set of looping directions, \[ \mathcal{L}_Σ = \{ (x,…
On any compact Riemannian manifold $(M, g)$ of dimension $n$, the $L^2$-normalized eigenfunctions $\{ϕ_λ\}$ satisfy $||ϕ_λ||_{\infty} \leq C λ^{\frac{n-1}{2}}$ where $-Δϕ_λ = λ^2 ϕ_λ.$ The bound is sharp in the class of all $(M, g)$ since it is obtained by zonal spherical harmonics on the standard $n$-sphere $S^n$. But of course, it is not sharp for many Riemannian manifolds, e.g. flat tori $\R^n/Γ$. We say that $S^n$, but not $\R^n/Γ$, is a Riemannian manifold with maximal eigenfunction growth. The problem which motivates this paper is to determine the $(M, g)$ with maximal eigenfunction grow…
Quantum ergodic restriction (QER) is the problem of finding conditions on a hypersurface $H$ so that restrictions $ϕ_j |_H$ to $H$ of $Δ$-eigenfunctions of Riemannian manifolds $(M, g)$ with ergodic geodesic flow are quantum ergodic on $H$. We prove two kinds of results: First (i) for any smooth hypersurface $H$, the Cauchy data $(ϕ_j|H, \partial ϕ_j|H)$ is quantum ergodic if the Dirichlet and Neumann data are weighted appropriately. Secondly (ii) we give conditions on $H$ so that the Dirichlet (or Neumann) data is individually quantum ergodic. The condition involves the almost nowhere equalit…
We show that one can obtain improved $L^4$ geodesic restriction estimates for eigenfunctions on compact Riemannian surfaces with nonpositive curvature. We achieve this by adapting Sogge's strategy in proving improved critical $L^p$ estimates. We first combine the improved $L^2$ restriction estimate of Blair and Sogge and the classical improved $L^\infty$ estimate of Bérard to obtain an improved weak-type $L^4$ restriction estimate. We then upgrade this weak estimate to a strong one by using the improved Lorentz space estimate of Bak and Seeger. This estimate improves the $L^4$ restriction esti…
This study focuses on the exact observability of a non-classical Euler-Bernoulli micro-beam equation. This non-classical model was derived based on the strain gradient elasticity theory, which is intended to explain the phenomenon of size effect at the micron scale. Spectral properties of the corresponding state operator are studied; an asymptotic expression for eigenvalues is calculated, and eigenfunctions are analyzed in order to check the necessary conditions for the exact observability of the system. By examining the eigenfunctions, it is shown that among non-collocated boundary outputs, o…
We extend a result of the second author \cite[Theorem 1.1]{soggekaknik} to dimensions $d \geq 3$ which relates the size of $L^p$-norms of eigenfunctions for $2<p<\frac{2(d+1)}{d-1}$ to the amount of $L^2$-mass in shrinking tubes about unit-length geodesics. The proof uses bilinear oscillatory integral estimates of Lee \cite{leebilinear} and a variable coefficient variant of an "$\veps$ removal lemma" of Tao and Vargas \cite{tv1}. We also use Hörmander's \cite{HorOsc} $L^2$ oscillatory integral theorem and the Cartan-Hadamard theorem to show that, under the assumption of nonpositive curvature, …
The goal of this article is to draw new applications of small scale quantum ergodicity in nodal sets of eigenfunctions. We show that if quantum ergodicity holds on balls of shrinking radius $r(λ) \to 0$, then one can achieve improvements on the recent upper bounds of Logunov and Logunov-Malinnikova on the size of nodal sets, according to a certain power of $r(λ)$. We also show that the order of vanishing results of Donnelly-Fefferman and Dong can be improved. Since by the results of Han and Hezari-Rivière small scale QE holds on negatively curved manifolds at logarithmically shrinking rates, w…
Let $(M,g)$ be an $n$-dimensional compact boudaryless Riemannian manifold with nonpositive sectional curvature, then our conclusion is that we can give improved estimates for the $L^p$ norms of the restrictions of eigenfunctions to smooth submanifolds of dimension $k$, for $p>\dfrac{2n}{n-1}$ when $k=n-1$ and $p>2$ when $k\leq n-2$, compared to the general results of Burq, Gérard and Tzvetkov \cite{burq}. Earlier, Bérard \cite{Berard} gave the same improvement for the case when $p=\infty$, for compact Riemannian manifolds without conjugate points for $n=2$, or with nonpositive sectional curvat…
This article contains a generalization of the authors' results on numbers of nodal points of eigenfunctions on "good curves" in analytic plane domains (arXiv:0710.0101). The term `good' means that the $L^2$ norms of restrictions of eigenfunctions of eigenvalue $λ^2$ to the curve are bounded below by $e^{- C λ}$. In this article, the result is generalized to all real analytic Riemannian manifolds $(M, g)$ of any dimension $m$ without boundary. Moreover, a similar lower bound is given for the Hausdorff $m-2$ measure of the intersection of the nodal set with a good real analytic hypersurface. Mos…
