arXiv research

Study bounds the measure of zero sets of Neumann Laplace eigenfunctions.

Study critical points of Laplace eigenfunctions in polygons.

Study heat profiles and eigenfunctions using Brownian motion.

The study proves constant-curvature analogues of hot spots conjecture for triangles.

Hot spots conjecture proven for small eigenvalue domains.

Let $M$ be a compact, connected Riemannian manifold whose Riemannian volume measure is denoted by $σ$. Let $f: M \rightarrow \mathbb{R}$ be a non-constant eigenfunction of the Laplacian. The random wave conjecture suggests that in certain situations, the value distribution of $f$ under $σ$ is approximately Gaussian. Wr…

Generalizing Courant's nodal domain theorem, the "Extended Courant property" is the statement that a linear combination of the first $n$ eigenfunctions has at most $n$ nodal domains. A related question is to estimate the number of connected components of the (super) level sets of a Neumann eigenfunction $u$. Indeed, in…

Two Riemannian manifolds are said to be isospectral if the associated Laplace-Belttrami operators have the same eigenvalue spectrum. If the manifolds have boundary, one specifies DIrichlet or Neumann isospectrality depending on the boundary conditions imposed on the eigenfunctions. We construct continuous families of (…

Abstract: Shows nonexistence of Courant-type nodal domain bounds for eigenfunctions of Dirichlet-to-Neumann operator.

Study small perturbations on low energy Laplace eigenfunctions.

The study uses heat flow to analyze properties of Laplace eigenfunctions on manifolds and domains.

We establish integral formulas and sharp two-sided bounds for the Ricci curvature, mean curvature and second fundamental form on a Riemannian manifold with boundary. As applications, sharp gradient and Hessian estimates are derived for the Dirichlet and Neumann eigenfunctions.

New upper bound found for nodal sets of Laplace eigenfunctions.

The study bounds quantum eigenfunctions on complex manifolds.

Study of loops in sums of Laplace eigenfunctions on surfaces.

The paper bounds Cheeger ratios of eigenfunctions and their level sets.

New method for minimal submanifolds in spheres using eigenfunctions.

We consider the eigenfunctions of the Laplace operator $Δ$ on a compact Riemannian manifold of dimension $n$. For $M$ homogeneous with irreducible isotropy representation and for a fixed eigenvalue of $Δ$ we find the average number of common zeros of $n$ eigenfunctions. For this we compute the volume of the image of $M…

Geometrically connects Laplace eigenfunctions to Borel-Weil theory on symmetric spaces.

Study on hyperbolic manifolds finds measures of Laplace eigenfunctions restricted to cosphere bundles.

New contractible domains on half-sphere with constant boundary Laplacian eigenfunctions.

Eigenfunction gradients on curved spaces imply rigid structure.

The study sharpens local Bernstein estimates for Laplace eigenfunctions on compact manifolds.

The paper proves inequalities linking Wasserstein distances and eigenfunctions in RCD(K,∞) spaces.

We establish the existence of analytic curves of eigenvalues for the Laplace-Neumann operator through an analytic variation of the metric of a compact Riemannian manifold $M$ with boundary by means of a new approach rather than Kato's method for unbounded operators. We obtain an expression for the derivative of the cur…

Let $Ω\subset R^2$ be a bounded piecewise smooth domain and $φ_λ$ be a Neumann (or Dirichlet) eigenfunction with eigenvalue $λ^2$ and nodal set ${ N}_{φ_λ} = {x \in Ω; φ_λ(x) = 0}.$ Let $H \subset Ω$ be an interior $C^ω$ curve. Consider the intersection number $n(λ,H):= \# (H \cap N_{φ_λ} ).$ We first prove that fo…

Generalizes crystallographic properties to all dimensions.

The paper calculates Morse indices and nullities for embedded networks on spheres.

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 prove the existence of a countable family of Delaunay type domains Ω_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 in M^n) for which the first eigenfunction of the Laplace-Beltrami operator with zero …

Researchers prove hot spots conjecture for Gaussian spaces.

Improved Kuznecov remainder estimates for generic metrics.

Trapezoids uniquely identified by their Dirichlet Laplace spectrum.

Anisotropic metric on manifolds uniquely determined by boundary data.

Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. Based on a combination of analytical and numerical results, we conjecture…

A compact Riemannian manifold may be immersed into Euclidean space by using high frequency Laplace eigenfunctions. We study the geometry of the manifold viewed as a metric space endowed with the distance function from the ambient Euclidean space. As an application we give a new proof of a result of Burq-Lebeau and othe…

Sharp bounds for eigenfunctions on product spaces restricted to submanifolds.

Let X be a manifold equipped with a complete Riemannian metric of constant negative curvature and finite volume. We demonstrate the finiteness of the collection of totally geodesic immersed hypersurfaces in X that lie in the zero-level set of some Laplace eigenfunction. For surfaces, we show that the number can be boun…

Let $Ω$ be an open, bounded domain in the plane with connected and smooth boundary, and $ω$ an eigenfunction of the Neumann Laplacian corresponding to some Neumann eigenvalue $μ> 0$. If the boundary value of $ω$ is a nonzero constant along the boundary, denoting $0 = μ_1(Ω) < μ_2(Ω) <= ...$ the set of all Neumann eigen…

In this paper we present a proof of a Neumann type maximum principle for the Laplace operator on compact Riemannian manifolds. A key p oint is the simple geometric nature of the constant in the a priori estimate of this maximum principle. In particular, this maximum principle can be applied to manifolds with Ricci curv…

In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures correspondin…

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.

This is a review of old and new results and methods related to the Yau conjecture on the zero set of Laplace eigenfunctions. The review accompanies two lectures given at the conference CDM 2018. We discuss the works of Donnelly and Fefferman including their solution of the conjecture in the case of real-analytic Rieman…

Infinite volume requires no atoms at the bottom of the spectrum for certain groups.

The study finds conditions for minimal spheres into ellipsoids using eigenfunctions.

The study finds multiple maxima for eigenfunctions on positively curved spheres.

It is proved that if a Paley-Wiener family of eigenfunctions of the Laplace operator in $\mathbb R^3$ vanishes on a real analytically ruled two-dimensional surface $S \subset \mathbb R^3$ then $S$ is a union of cones, each of which is contained in a translate of the zero set of a nonzero harmonic homogeneous polynomial…

Study covariant derivatives of eigenfunctions on curved spaces, proving they are scalar multiples of the functions.