Study critical points of Laplace eigenfunctions in polygons.
problem Characterize critical points of Laplace eigenfunctions in polygonal domains.
method Analyze components of the critical set with codimension 1.
result For simply connected polygons, if a second Neumann eigenfunction has infinitely many critical points, the polygon must be a rectangle.
Study bounds the measure of zero sets of Neumann Laplace eigenfunctions.
problem Bounding the measure of zero sets of Neumann Laplace eigenfunctions.
method Analyzes nodal sets in domains with C 1 , 1 C^{1,1} C 1 , 1 boundaries. result Hausdorff measure bound of nodal sets is C λ C \sqrt{\lambda} C λ . Study small perturbations on low energy Laplace eigenfunctions.
problem Understanding small changes in low energy Laplace eigenfunctions.
method Investigates nodal geometry and topology, focusing on low frequency regimes and small perturbations.
result Highlight interesting aspects of spectral theory and nodal phenomena tied to ground state/low energy eigenfunctions.
The study uses heat flow to analyze properties of Laplace eigenfunctions on manifolds and domains.
problem Analyzing mass concentration and nodal domains of Laplace eigenfunctions.
method Heat diffusion technique to study eigenfunctions and their nodal sets.
result Discovers new insights into the decay and behavior of Laplace eigenfunctions.
Study finds average number of common zeros for eigenfunctions on a compact manifold.
problem Determining the average number of common zeros for eigenfunctions on a compact manifold.
method Computing the volume of the image of the manifold under an equivariant immersion into a sphere.
result Average number of common zeros for n n n eigenfunctions is computed. New upper bound found for nodal sets of Laplace eigenfunctions.
problem Finding the maximum area of nodal sets for Laplace eigenfunctions.
method Analyzing the ( n − 1 ) (n-1) ( n − 1 ) -dimensional Hausdorff measure of zero sets of eigenfunctions. result The sharp upper bound for the area of nodal sets is C ( Ω ) λ C(Ω)\sqrtλ C ( Ω ) λ . The study bounds quantum eigenfunctions on complex manifolds.
problem Restricting quantum eigenfunctions on complex manifolds.
method Analytic continuation and FBI transform for Laplace eigenfunctions.
result Upper and lower L 2 L^2 L 2 bounds for eigenfunctions. Review of Yau's conjecture on zero sets of Laplace eigenfunctions.
problem Yau's conjecture on zero sets of Laplace eigenfunctions.
method Discussion of old and new results and methods related to the conjecture, including solutions and new results in smooth settings.
result Discussion of Donnelly and Fefferman's solution of the conjecture in the real-analytic Riemannian manifold case and new results in the smooth setting.
Study of loops in sums of Laplace eigenfunctions on surfaces.
problem Uniform bound for the number of nested loops in sums of Laplace eigenfunctions.
method Real-analytic category analysis and biharmonic function construction.
result Uniform bound for the number of rooted double nests in terms of surface, root, and spectral cutoff.
The paper bounds Cheeger ratios of eigenfunctions and their level sets.
problem Understanding geometric features of Riemannian manifolds through eigenfunctions.
method Constructive upper bounds on Cheeger constants using eigenvalues and eigenfunctions.
result Upper bounds on Cheeger ratios of eigenfunction level sets and their superlevel sets.
New method for minimal submanifolds in spheres using eigenfunctions.
problem Constructing minimal submanifolds in spheres.
method Eigenfunctions of the Laplace-Beltrami operator to describe minimal submanifolds.
result New proof of ( λ , μ ) (λ, μ) ( λ , μ ) -eigenfunctions on spheres. The study improves estimates of zero sets of Laplace eigenfunctions in 2D and 3D.
problem Estimating the Hausdorff measure of zero sets of Laplace eigenfunctions.
method Combination of Donnelly-Fefferman estimate and combinatorial argument.
result Improved estimates for Hausdorff measure in 2D and 3D.
Geometrically connects Laplace eigenfunctions to Borel-Weil theory on symmetric spaces.
problem Understanding the spectral properties of Laplace-Beltrami operators on Riemannian symmetric spaces.
method Using symplectic geometry and geometric quantization, associating flag manifolds to symmetric spaces and relating their Satake diagrams.
result Harmonic polynomials on flag manifolds induce all eigenfunctions on symmetric spaces.
Eigenfunctions of Laplacian form knots in 3-manifolds.
problem Creating metrics with Laplacian eigenfunctions that form specific knots.
method Constructing a Riemannian metric on a 3-manifold M such that the first nontrivial eigenfunction of the Laplacian has a nodal set shaped like a given knot.
result Existence of a Riemannian metric on a 3-manifold M with a Laplacian eigenfunction whose nodal set is a given knot.
The study reveals properties of ruled surfaces in 3D space related to Laplace eigenfunctions and spherical mean transforms.
problem Characterizing ruled surfaces in 3D space where Laplace eigenfunctions vanish.
method Analyzing the zero sets of Laplace eigenfunctions and their relation to ruled surfaces.
result Ruled surfaces in 3D space are composed of cones whose zero sets are related to harmonic polynomials.
Study heat profiles and eigenfunctions using Brownian motion.
problem Investigate heat profiles and eigenfunctions of Laplace equations.
method Probabilistic tools based on Brownian motion and Feynman-Kac formulae.
result Supremum norm bounds for ground state Dirichlet eigenfunctions and comparison of maximum temperatures.
Study on hyperbolic manifolds finds measures of Laplace eigenfunctions restricted to cosphere bundles.
problem Analyzing semiclassical measures on hyperbolic manifolds.
method Adapting Dyatlov and Jin's argument to higher dimensions and using Ratner theory.
result Semiclassical measures' support contains the cosphere bundle of a compact totally geodesic submanifold.
Eigenfunction value distribution shows unimodal density with maximum at zero.
problem Understanding the value distribution of Laplace eigenfunctions.
method Analyzing the measure μ μ μ whose density is ∣ a b l a f ∣ 2 |
abla f|^2 ∣ ab l a f ∣ 2 and proving a monotonicity formula. result Eigenfunction value distribution under μ μ μ is unimodal with maximum at zero. The study bounds the Hausdorff measure of nodal sets of Laplace eigenfunctions.
problem Bounding the Hausdorff measure of nodal sets of Laplace eigenfunctions.
method Propagation of smallness technique for elliptic PDE solutions.
result Polynomial upper estimates of the Hausdorff measure for nodal sets.
Proves Nadirashvili's conjecture and Yau's lower bound for Laplace eigenfunctions.
problem Proving bounds on nodal sets of Laplace eigenfunctions.
method Analytic proof on smooth manifolds.
result Establishes lower bounds on the volume of zero sets of Laplace eigenfunctions.
The study sharpens local Bernstein estimates for Laplace eigenfunctions on compact manifolds.
problem Understanding local growth properties of Laplace eigenfunctions on compact Riemannian manifolds.
method Refined Donnelly-Fefferman method based on L 2 L^{2} L 2 --Carleman estimates, combined with elliptic regularity and patching of local Carleman estimates. result Almost sharp local L p L^{p} L p --Bernstein inequalities for p ∈ [ 1 , ∞ ] p\in[1,\infty] p ∈ [ 1 , ∞ ] . Study geodesics and nodal sets on hyperbolic surfaces, proving finiteness results.
problem Understanding the distribution of geodesics and nodal sets on hyperbolic manifolds.
method Analyzing Laplace eigenfunctions on hyperbolic manifolds, focusing on geodesic and nodal properties.
result Finiteness of geodesic hypersurfaces in zero-level sets of Laplace eigenfunctions.
The paper proves inequalities linking Wasserstein distances and eigenfunctions in RCD(K,∞) spaces.
problem Estimating Wasserstein distances and their bounds in RCD(K,∞) spaces.
method Similar techniques used to prove inequalities involving p p p -Wasserstein distances and Laplace eigenfunctions. result Proves a conjectured lower bound on p p p -Wasserstein distance between positive and negative parts of Laplace eigenfunctions. Eigenfunctions on manifolds concentrate near nodal sets.
problem Understanding concentration of eigenfunctions on manifolds.
method Volume measure analysis and Colding-Minicozzi method.
result Exponential concentration of volume measure around nodal sets.
Generalizes crystallographic properties to all dimensions.
problem Analytic eigenfunctions in crystallographic groups.
method Algebraic, geometric, and analytic proofs.
result Equivalent conditions for real analytic eigenfunctions in crystallographic polytopes.
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.
Improved Kuznecov remainder estimates for generic metrics.
problem Estimating period integrals of Laplace eigenfunctions on manifolds.
method Two-term asymptotic expansion and elimination of oscillatory second term.
result Improved remainder estimates for Baire-generic metrics.
The study proves constant-curvature analogues of hot spots conjecture for triangles.
problem Proving the hot spots conjecture in constant curvature domains.
method Analyzing geodesic triangles of constant negative curvature and using Killing fields.
result First mixed Dirichlet-Neumann Laplace eigenfunctions have no non-vertex critical points in constant curvature triangles.
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.
problem Establishing bounds for eigenfunctions on product spaces.
method Combining asymptotics of Jacobi polynomials and positivity of Fourier coefficients of spherical functions.
result Sharp L p L^p L p bounds for eigenfunctions on products of rank-one symmetric spaces. 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 L^2 L 2 eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with Q \mathbb{Q} Q -rank one if the corresponding eigenvalues lie below the continuous part of the L 2 L^2 L 2 spectrum. Furthermore, we use these bounds in order to obtain some results concerning the L p L^p L p spectrum.
Eigenfunctions of collapsing Einstein manifolds are almost constant along fibers.
problem Eigenfunctions of collapsing Einstein manifolds.
method Cheeger-Colding's almost splitting theorem and harmonic almost splitting map.
result Eigenfunctions are almost constant along fibers in the L 2 L^2 L 2 -average sense. Infinite volume requires no atoms at the bottom of the spectrum for certain groups.
problem Determining conditions for infinite volume in certain algebraic groups.
method Analyzing the spectral properties of Laplace operators on symmetric spaces.
result The bottom of the L 2 L^2 L 2 -spectrum being an atom is necessary and sufficient for finite volume. The study finds conditions for minimal spheres into ellipsoids using eigenfunctions.
problem Conditions for branched minimal immersions of spheres into ellipsoids to be embedded.
method Using eigenfunctions with respect to a critical metric, the study provides sufficient conditions for embeddings and constructions of non-planar minimal spheres.
result Conditions for embeddings and constructions of non-planar minimal spheres using eigenfunctions.
The study finds multiple maxima for eigenfunctions on positively curved spheres.
problem Finding multiple non-degenerate maxima for eigenfunctions on positively curved surfaces.
method Proving the existence of a smooth closed Riemannian surface with positive Gaussian curvature and specific eigenfunction properties.
result There exist surfaces with at least m distinct non-degenerate local maxima for the first nonzero eigenfunction.
The study examines eigenfunctions on lens spaces using harmonic-counting measures.
problem Understanding the asymptotic behavior of eigenfunctions on lens spaces.
method Introduced harmonic-counting measures and applied them to lens spaces.
result Determined the asymptotic behavior of eigenfunctions associated with lattice points.
The paper bounds eigenvalues and integrals of eigenfunctions on hyperbolic manifolds.
problem Eigenvalues and integrals of eigenfunctions on compact hyperbolic manifolds.
method Spectral decompositions and consistency conditions derived from quadruple overlap integrals.
result Upper bounds on Laplacian eigenvalues and triple overlap integrals.
Study covariant derivatives of eigenfunctions on curved spaces, proving they are scalar multiples of the functions.
problem Understanding covariant derivatives of eigenfunctions on curved spaces.
method Analyzing the Laplace-Beltrami operator on Riemannian manifolds with constant curvature.
result Covariant derivatives of eigenfunctions are scalar multiples of the functions, and these scalars are polynomials in the eigenvalue.
Spectral calculus simplifies manifold learning with eigenfunctions.
problem Learning manifold structures from data.
method Reformulate exterior calculus using Laplacian eigenfunctions.
result Spectral convergence to true exterior calculus.
Study on variance of Laplace eigenfunctions on manifolds.
problem Investigating the variance of Laplace eigenfunctions on compact manifolds.
method Combining Kac-Rice formula, Wiener-Itô chaos decompositions, and pointwise Weyl law analysis.
result Established a quantitative bound for the fluctuations of nodal volumes, improving existing results.
Self-focal points on ellipsoids of dimension 3 or higher are rare.
problem Existence of self-focal points on Riemannian manifolds of dimension 3 or higher.
method Analyzing geodesics and umbilic points on ellipsoids of various dimensions.
result Ellipsoids of dimension 3 or higher with at least 4 distinct axes have no self-focal points.
Estimates average number of common zeros of Laplacian eigenfunctions on manifolds.
problem Estimating the average number of common zeros of Laplacian eigenfunctions on compact Riemannian manifolds.
method Application of Crofton's formula for the sphere.
result Proves an estimate for the average number of common zeros of eigenfunctions, showing it does not exceed a specific expression.
This purpose of this write-up is to share an idea for accurate computation of Laplace eigenvalues on a broad class of smooth domains. We represent the eigenfunction u u u as a linear combination of eigenfunctions corresponding to the common eigenvalue ρ 2 ρ^{2} ρ 2 :\EQN{6}{1}{}{0}{\RD{\CELL{u(r,θ) =\sum_{n=0}^{N}P_{n}J_{n}(ρ) …
New Finslerian Ressiner-Nordstrom spacetime with constant flag curvature.
problem Exploring a new spacetime model with constant flag curvature.
method Derived Finslerian Ressiner-Nordstrom solution and analyzed its properties.
result The solution differs from Ressiner-Nordstrom metric only in two dimensional subspace with constant flag curvature.
The study examines surfaces in isotropic space with specific Gauss map properties.
problem Understanding surfaces in simply isotropic space with degenerate metric.
method Investigates surfaces with Gauss map coordinates as eigenfunctions of the Laplace-Beltrami operator for minimal and parabolic normals.
result Identifies surfaces characterized by eigenfunction properties of the Gauss map.
Study meromorphic open-string vertex algebras and modules over Riemannian manifolds.
problem Characterize meromorphic open-string vertex algebras and their modules over Riemannian manifolds.
method Explicitly determine bases for meromorphic open-string vertex algebras and their modules, using parallel tensors and eigenfunctions of the Laplace-Beltrami operator.
result Every irreducible module of a specific type is completely reducible if every composition factor is generated by eigenfunctions of eigenvalue p ( p − 1 ) K p(p-1)K p ( p − 1 ) K for some p ∈ Z + p\in \mathbb{Z}_+ p ∈ Z + . Study on eigenfunctions on sphere configurations, proving non-existence and construction of critical eigensections.
problem Existence and rigidity of critical Z2 eigenvalues on sphere configurations.
method Algebraic identities and finite group representation theory.
result Construction of infinitely many configurations admitting critical eigensections and proof of deformation rigidity of Taubes-Wu tetrahedral eigensections.