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 estimate the volume of superlevel sets of Laplace-Beltrami eigenfunctions on a compact Riemannian manifold. The proof uses the Green's function representation and the Bathtub principle. As an application, we obtain upper bounds on the distribution of the extrema of a Laplace-Beltrami eigenfunction over its nodal domains. Such bounds have been previously proved by L. Polterovich and M. Sodin in the case of compact surfaces. Our techniques allow to generalize these results to arbitrary dimensions. We also discuss a different approach to the problem based on reverse Hölder inequalities due to …
Eigenfunctions of the Laplace-Beltrami operator on a hyperboloid are studied in the spirit of the treatment of the spherical harmonics by Stein and Weiss. As a special case, a simple self-contained proof of Laplace's integral for a Legendre function is obtained.
In this note we analyse $L^{p}$ estimates for Laplacian eigenfunctions and quasimodes and their associated sharp examples. In particular we use previously determined estimates to produce a new set of estimates for restriction to thickened neighbourhoods of submanifolds. In addition we produce a family flat model quasimode examples that can be used to determine sharpness of estimates on Laplacian eigenfunctions restricted to subsets. For each quasimode in the family we show that there is a corresponding spherical harmonic that displays the same growth properties. Therefore it is enough to check…
Let $e(x,y,ł)$ be the spectral function and $χ_ł$ the unit band spectral projection operator, with respect to the Laplace-Beltrami operator $\D_M$ on a closed Riemannian manifold $M$. We firstly review the one-term asymptotic formula of $e(x,x,ł)$ as $ł\to\infty$ by H{\" o}rmander (1968) and the one of $\p^\al_x\p^\bt_y e(x,y,ł)|_{x=y}$ as $ł\to\infty$ in a geodesic normal coordinate chart by the author (2004) and the sharp asymptotic estimates from above of the mapping norm $\|χ_ł\|_{L_2\to L_p}$ ($2\leq p\leq\infty$) by Sogge (1988 $&$ 1989) and of the mapping norm $\|χ_ł\|_{L_2\to {\rm Sobo…
We study $L^p$ bounds on spectral projections for the Laplace operator on compact Riemannian manifolds, restricted to small frequency dependent neighborhoods of submanifolds. In particular, if $λ$ is a frequency and the size of the neigborhood is $\mathcal{O}(λ^{-δ})$, then new sharp estimates are established when $δ\ge 1$, while for $0\le δ\le 1/2$, Sogge's estimates turn out to be optimal. In the intermediate region $1/2<δ<1$, we sometimes get sharp estimates as well. Our arguments follow closely a recent work by Burq and Zuily.
We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.
We describe a new approach to understanding averages of high energy Laplace eigenfunctions, $u_h$, over submanifolds, $$ \Big|\int _H u_hdσ_H\Big| $$ where $H\subset M$ is a submanifold and $σ_H$ the induced by the Riemannian metric on $M$. This approach can be applied uniformly to submanifolds of codimension $1\leq k\leq n$ and in particular, gives a new approach to understanding $\|u_h\|_{L^\infty(M)}$. The method, developed in the author's recent work together with Y. Canzani and J. Toth, relies on estimating averages by the behavior of $u_h$ microlocally near the conormal bundle to $H$. By…
We obtain restrictions on the persistence barcodes of Laplace-Beltrami eigenfunctions and their linear combinations on compact surfaces with Riemannian metrics. Some applications to uniform approximation by linear combinations of Laplace eigenfunctions are also discussed.
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 …
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…
We consider linear combinations of eigenfunctions of the Laplace-Beltrami operator on a compact Riemannian manifold $(M,g)$ and investigate a density property of their zero sets. More precisely, let $f=\sum_{k=1}^m a_k ϕ_{λ_{j_k}}$, where $-Δ_gϕ_λ=λϕ_λ$. Denoting by $Z_f$ the zero-set of $f$, we show that for any $x\in M$, $dist(x,Z_f)\leq C(m)λ_{j_1}^{-1/2}$. The proof is based on a new integral Harnack-type estimate for positive solutions of higher order elliptic PDEs.
Consider an $L^2$-normalized Laplace-Beltrami eigenfunction $e_λ$ on a compact, boundary-less Riemannian manifold with $Δe_λ= -λ^2 e_λ$. We study eigenfunction triple products \[ \langle e_λe_μ, e_ν\rangle = \int e_λe_μ\overline{e_ν} \, dV. \] We show the overall $\ell^2$-concentration of these triple products is determined by the measure of some set of configurations of triangles with side lengths equal to the frequencies $λ,μ,$ and $ν$. A rapidly vanishing proportion of this mass lies in the `classically forbidden' regime where $λ, μ,$ and $ν$ fail to satisfy the triangle inequality. As a co…
Let $e_ł(x)$ be an eigenfunction with respect to the Laplace-Beltrami operator $Δ_M$ on a compact Riemannian manifold $M$ without boundary: $Δ_M e_ł=ł^2 e_ł$. We show the following gradient estimate of $e_ł$: for every $ł\geq 1$, there holds $ł\|e_ł\|_\infty/C\leq \|\nabla e_ł\|_\infty\leq Cł\|e_ł\|_\infty$, where $C$ is a positive constant depending only on $M$.
We show that, in round spheres of dimension $n\geq3$, for any given collection of codimension 2 smooth submanifolds $\mathfrak{S}:=\{Σ_1,...,Σ_N\}$ of arbitrarily complicated topology ($N$ being the complex dimension of the spinor bundle), there is always an eigenfunction $ψ=(ψ_1,...,ψ_N)$ of the Dirac operator such that each submanifold $Σ_a$, modulo ambient diffeomorphism, is a structurally stable nodal set of the spinor component $ψ_a$. The result holds for any choice of trivialization of the spinor bundle. The emergence of these structures takes place at small scales and sufficiently high …
We prove pointwise bounds for $L^2$ eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with $\mathbb{Q}$-rank one if the corresponding eigenvalues lie below the continuous part of the $L^2$ spectrum. Furthermore, we use these bounds in order to obtain some results concerning the $L^p$ spectrum.
We study the relationship between $L^\infty$ growth of eigenfunctions and their $L^2$ concentration as measured by defect measures. In particular, we show that scarring in the sense of concentration of defect measure on certain submanifolds is incompatible with maximal $L^\infty$ growth. In addition, we show that a defect measure which is too diffuse, such as the Liouville measure, is also incompatible with maximal eigenfunction growth.
We give a proof of the Donnelly-Fefferman growth bound of Laplace-Beltrami eigenfunctions which is probably the easiest and the most elementary one. Our proof also gives new quantitative geometric estimates in terms of curvature bounds which improve and simplify previous work by Garofalo and Lin. The proof is based on a convexity property of harmonic functions on curved manifolds, generalizing Agmon's Theorem on a convexity property of harmonic functions in R^n.
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…
The Baran metric $δ_E$ is a Finsler metric on the interior of $E\subset \R^n$ arising from Pluripotential Theory. We consider the few instances, namely $E$ being the ball, the simplex, or the sphere, where $δ_E$ is known to be Riemaniann and we prove that the eigenfunctions of the associated Laplace Beltrami operator (with no boundary conditions) are the orthogonal polynomials with respect to the pluripotential equilibrium measure $μ_E$ of $E.$ We conjecture that this may hold in a wider generality. The considered differential operators have been already introduced in the framework of orthogon…
Let M be a closed Riemannian manifold of dimension n. Let f be an eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue λ. We show that the volume of {f>0} inside any ball B whose center lies on {f=0} is > C|B|/λ^n. We apply this result to prove that each nodal domain contains a ball of radius > C/λ^n.
Let $(M,g)$ be a compact Riemannian surface. Consider a family of $L^2$ normalized Laplace-Beltrami eigenfunctions, written in the semiclassical form $-h_j^2Δ_g ϕ_{h_j} = ϕ_{h_j}$, whose eigenvalues satisfy $h h_j^{-1} \in (1, 1 + hD]$ for $D>0$ a large enough constant. Let $\mathbf{P}_h$ be a uniform probability measure on the $L^2$ unit-sphere $S_h$ of this cluster of eigenfunctions and take $u \in S_h$. Given a closed curve $γ\subset M$, there exists $C_{1}(γ, M), C_{2}(γ, M) > 0$ and $h_0>0$ such that for all $h \in (0, h_0],$ \begin{equation*} C_1 h^{1/2} \leq \mathbf{E}_{h} \bigg[ \big| …
Let $\{e_j\}$ be an orthonormal basis of Laplace eigenfunctions of a compact Riemannian manifold $(M,g)$. Let $H \subset M$ be a submanifold and let $\{ψ_k\}$ be an orthonormal basis of Laplace eigenfunctions of $H$ with the induced metric. We obtain joint asymptotics for the Fourier coefficients \[ \langle γ_H e_j, ψ_k \rangle_{L^2(H)} = \int_H e_j \overline ψ_k \, dV_H, \] of restrictions $γ_H e_j$ of $e_j$ to $H$. In particular, we obtain asymptotics for the sums of the norm-squares of the Fourier coefficients over the joint spectrum $\{(μ_k, λ_j)\}_{j,k - 0}^{\infty}$ of the (square roots …
The eigenvalues of the Laplace-Beltrami operator and the integrals of products of eigenfunctions must satisfy certain consistency conditions on compact Riemannian manifolds. These consistency conditions are derived by using spectral decompositions to write quadruple overlap integrals in terms of products of triple overlap integrals in multiple ways. In this paper, we show how these consistency conditions imply bounds on the Laplacian eigenvalues and triple overlap integrals of closed hyperbolic manifolds, in analogy to the conformal bootstrap bounds on conformal field theories. We find an uppe…
The Laplace-Beltrami operator of a smooth Riemannian manifold is determined by the Riemannian metric. Conversely, the heat kernel constructed from its eigenvalues and eigenfunctions determines the Riemannian metric. This work proves the analogy on Euclidean polyhedral surfaces (triangle meshes), that the discrete Laplace-Beltrami operator and the discrete Riemannian metric (unique up to a scaling) are mutually determined by each other. Given an Euclidean polyhedral surface, its Riemannian metric is represented as edge lengths, satisfying triangle inequalities on all faces. The Laplace-Beltrami…
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 i…
The Laplace-Beltrami eigenfunctions on a compact Riemannian manifold $M$ whose geodesic billiard flow has mixed character have been conjectured by Percival to split into two complementary families, with all semiclassical mass supported in the completely integrable and ergodic regions of phase space respectively. In this paper, we consider the Dirichlet Laplacian on a family of mushroom billiards $M_t$ parametrised by the length $t\in(0,2]$ of their rectangular part. We prove that $M_t$ satisfies Percival's conjecture for almost all $t\in(0,2]$, hence providing the first example of a billiard k…
We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eigenfunctions. In particular in both cases we get a Weyl's law with leading term $E\log E$. We then study the drastic effect of Aharonov-Bohm magnetic potentials on the spectral properties. Other generalised Riemannian structures including conic and anti-conic type manifolds are als…
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 $ϕ_λ(x)$…
University of Minnesota（明尼苏达大学）；University of Wisconsin-Madison（威斯康星大学麦迪逊分校）
In this paper we improve the spectral convergence rates for graph-based approximations of Laplace-Beltrami operators constructed from random data. We utilize regularity of the continuum eigenfunctions and strong pointwise consistency results to prove that spectral convergence rates are the same as the pointwise consistency rates for graph Laplacians. In particular, for an optimal choice of the graph connectivity $\varepsilon$, our results show that the eigenvalues and eigenvectors of the graph Laplacian converge to those of the Laplace-Beltrami operator at a rate of $O(n^{-1/(m+4)})$, up to lo…
