arXiv research

New method for minimal submanifolds in spheres using eigenfunctions.

Unified framework for complex-valued eigenfunctions on Riemannian symmetric spaces.

Study critical points of Laplace eigenfunctions in polygons.

Establishes log-concavity estimates for convex domains' first Dirichlet eigenfunctions.

The study counts critical points of Steklov eigenfunctions on manifolds.

Study heat profiles and eigenfunctions using Brownian motion.

Eigenfunction gradients on curved spaces imply rigid structure.

New method creates minimal submanifolds using complex-valued eigenfunctions.

We study concentration phenomena of eigenfunctions of the Laplacian on closed Riemannian manifolds. We prove that the volume measure of a closed manifold concentrates around nodal sets of eigenfunctions exponentially. Applying the method of Colding and Minicozzi we also prove restricted exponential concentration inequa…

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

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

Let us fix two different radial eigenfunctions of a hyperbolic Laplacian and assume that both of them have the same value at the origin. Both eigenvalues can be complex numbers. The main goal of this paper is to estimate the lower bound for the interval (0,T], where these two eigenfunctions must assume different values…

The paper explores eigenfunctions of spherical conical metrics using harmonic maps to spheres.

Study small perturbations on low energy Laplace eigenfunctions.

Paper connects probability density cuts to graph theory eigenfunctions.

Gaussian processes (GPs) provide a nonparametric representation of functions. However, classical GP inference suffers from high computational cost and it is difficult to design nonstationary GP priors in practice. In this paper, we propose a sparse Gaussian process model, EigenGP, based on the Karhunen-Loeve (KL) expan…

Eigenfunction maxima inside high-d nodal domains.

We give an upper bound for the $(n-1)$-dimensional Hausdorff measure of the critical set of eigenfunctions of the Laplacian on compact analytic Riemannian manifolds. This is the analog of H. Donnely and C. Fefferman result on nodal set of eigenfunctions.

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…

In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the first eigenfunction of $p$-Laplacian, as $p\to 1,$ we ident…

We survey recent results related to the concentration of eigenfunctions. We also prove some new results concerning ball-concentration, as well as showing that eigenfunctions saturating lower bounds for $L^1$-norms must also, in a measure theoretical sense, have extreme concentration near a geodesic.

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 applicatio…

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

This study provides a new mathematical structure for Koopman eigenfunctions.

Improved bounds for eigenfunctions on hyperbolic surfaces found.

The study bounds quantum eigenfunctions on complex manifolds.

Machine learning finds Z/2 eigenfunctions on a sphere.

Study on $(λ,λ)$-eigenfunctions on compact manifolds, showing manifold properties and eigenfamily dimensions.

In this paper we consider the problem of prescribing the nodal set of low-energy eigenfunctions of the Laplacian. Our main result is that, given any separating closed hypersurface Σin a compact n-manifold M, there is a Riemannian metric on M such that the nodal set of its first nontrivial eigenfunction is Σ. We present…

We use persistent homology along with the eigenfunctions of the Laplacian to study similarity amongst triangulated 2-manifolds. Our method relies on studying the lower-star filtration induced by the eigenfunctions of the Laplacian. This gives us a shape descriptor that inherits the rich information encoded in the eigen…

Log-concavity of eigenfunctions on curved surfaces is proven, leading to fundamental gap estimates.

We prove lower bounds for the Hausdorff measure of nodal sets of eigenfunctions.

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

Study eigenfunctions of Laplacian on sphere with even point removals.

Unified framework for constructing kernels for transport equations and Koopman eigenfunctions.

Sharp bounds for eigenfunctions on product spaces restricted to submanifolds.

Paper proves super log-concavity of first eigenfunction for certain hyperbolic domains.

Study on eigenfunctions on sphere configurations, proving non-existence and construction of critical eigensections.

New upper bound found for nodal sets of Laplace eigenfunctions.

Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.

The paper finds new eigenfunctions for minimal immersions and their stability index.

Proposes a framework to extract ordered eigenfunctions from contextual kernels.

The paper studies eigenfunctions and nodal sets of the Witten-Laplacian.

New inequality for eigenfunctions on curved spaces.

Kernel method approximates Koopman operator eigenfunctions.

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

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…

Researchers create metrics for Laplacian eigenfunctions with specific zero sets.