New framework for studying eigenvalue functionals of metrics.
problem Understanding critical points of eigenvalue functionals.
method Clarke subdifferential theory to unify previous research.
result Unified understanding of critical metrics and new examples.
Paper derives second variation formula for eigenvalue functionals on surfaces.
problem Determine if a critical metric is a local maximizer for eigenvalue functionals.
method Derive second variation formula for critical metrics and apply to specific cases.
result Flat metric on non-rhombic torus cannot be a conformal maximizer for first eigenvalue.
The study classifies solutions to a specific eigenvalue problem and identifies the critical catenoid.
problem Eigenvalue problem on the sphere with boundary conditions.
method Classifying positive solutions as rotationally symmetric and analyzing boundary conditions.
result Characterization of the critical catenoid as the only embedded free boundary minimal annulus.
In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions for a metric to be a critical point of such a functional. We derive specific con…
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.
Hot spots conjecture proven for small eigenvalue domains.
problem Hot spots conjecture for hyperbolic planar domains with small eigenvalues.
method Proved a variant of Rauch's hot spots conjecture.
result Second Neumann Laplace eigenfunctions have no interior critical points on large convex domains.
In [LS], it is shown shown that the first eigenvalue of the Laplacian restricted to the space of invariant functions on a toric Kähler manifold (i.e. λ1T, the invariant first eigenvalue) is an unbounded function of the toric Kähler metric. In this note we show that, seen as a function on the space of toric…
Study on metrics maximizing eigenvalues of Paneitz operator on 4-manifolds.
problem Investigating metrics maximizing eigenvalues of Paneitz operator.
method Critical points of eigenvalues of Paneitz operator on Riemannian metrics with fixed volume.
result Critical metrics associated with extrinsic conformal-harmonic maps into round spheres.
Optimizing quantum graphs yields geodesic nets on surfaces.
problem Finding optimal quantum graphs for geodesic nets.
method Optimizing functionals from spectral theory to find geodesic nets.
result Critical metrics for eigenvalues give rise to geodesic nets.
The paper calculates critical configurations and Morse indices for polygons on circles or ellipses.
problem Finding critical configurations and their properties for polygons on circles or ellipses.
method Computing Morse indices and gradient vector fields for isolated critical points, relating to eigenvalue questions.
result Computed Morse indices and relationships to eigenvalue questions for polygons on circles or ellipses.
Abstract: Characterizes special Kähler manifolds with specific properties.
problem Characterizing Kähler manifolds with special properties.
method Analyzes properties of functions and gradients on manifolds.
result Characterizes manifolds supporting certain functions and gradients.
New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.
problem Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.
method Properties of magnetic geodesic flow and behavior at Mañé's critical energy level.
result Improved Cheeger constants and volume dependence in proofs.
Jakobson and Nadirashvili \cite{JN} constructed a sequence of eigenfunctions on T2 with a bounded number of critical points, answering in the negative the question raised by Yau \cite{Yau1} which asks that whether the number of the critical points of eigenfunctions for the Laplacian increases with the corresponding …
We show that an embedded minimal annulus Σ2⊂B3 which intersects ∂B3 orthogonally and is invariant under reflection through the coordinate planes is the critical catenoid. The proof uses nodal domain arguments and a characterization, due to Fraser and Schoen, of the critical catenoid as the unique…
New insights into matrix factorization show strict saddles have bounded eigenvalues.
problem Understanding the nature of critical points in matrix factorization.
method Analyzing orbits of critical points under the general linear group and identifying canonical points.
result Minimum eigenvalue of strict saddles is not uniformly bounded below zero.
The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.
problem Maximizing the first normalized Laplace-Beltrami eigenvalue on tori.
method Constructing equivariant harmonic maps to spheres and analyzing their properties.
result Rotationally symmetric critical metrics for the first eigenvalue are found and characterized.
Study improves regularity estimates for harmonic maps into ellipsoids.
problem Independence of regularity estimates on harmonic maps with varying target dimensions.
method Analyzes harmonic maps into ellipsoids, uses Palais-Smale sequences, and critical metrics.
result Enhanced regularity estimates for Laplace harmonic eigenmaps.
The present paper is devoted to geometric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains Ω of a Riemannian manifold (M,g). More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue μ2(Ω) with respect …
When a Riemannian manifold (M,g) is rotationally symmetric, the critical order of the lower bound of radial curvatures for the absence of eigenvalues of the Laplacian is equal to −r1, where r stands for the distance to the center point. In this paper, we shall perturb the Riemannian metric around a rota…
In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) k-th Steklov eigenvalue on the disk is not maximized for a smooth metric on the disk for k≥3. For k=1 the classical…
Study extends eigenvalue formulas to weighted manifolds and proves global rigidity theorems.
problem Eigenvalue formulas and rigidity theorems for weighted manifolds.
method Extends variational formulae to weighted manifolds, proving global rigidity theorems.
result Global rigidity theorems for critical domains in Gaussian half-space.
Given a simply connected compact generalized flag manifold M together with its invariant Kähler Einstein metric g, we investigate the functional given by the first eigenvalue of the Hodge Laplacian on smooth functions restricted to the space of invariant Kähler metrics. We give sufficient and necessary conditions so th…
Minimal surfaces in spheres have unique energy properties.
problem Characterizing minimal surfaces in spheres based on their energy index and eigenvalues.
method Analyzing the second variations of area and energy for minimal immersions.
result New bounds on the energy index and eigenvalues for minimal surfaces in spheres.
Study eigenvalues of Dirac operator on surfaces, proving existence and deriving inequalities.
problem Finding optimal bounds for Dirac eigenvalues on spin surfaces.
method Minimization problem within a fixed conformal class, focusing on surfaces.
result Derive isoperimetric inequalities for the Dirac operator on the sphere, complete conformal spectrum characterization.
Proves existence of non-planar minimal disks in ellipsoids.
problem Existence of non-planar minimal disks in ellipsoids.
method Optimization of Steklov eigenvalues and critical metrics.
result Existence of embedded non-planar free boundary minimal disks.
We prove the existence of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in any compact Riemannian manifold. This result generalizes a results of F. Pacard and the second author where the existence of a nondegenerate critical point of the scalar curvature of the …
The paper optimizes Dirac eigenvalues on surfaces and connects them to harmonic maps into complex projective spaces.
problem Optimizing the k-th positive Dirac eigenvalue on surfaces with fixed area and conformal class. method Connecting the problem to the maximization of Laplacian eigenvalues and using critical metrics for Dirac eigenvalues and harmonic maps into complex projective spaces.
result The first nonzero Dirac eigenvalue on a torus is minimized by the flat metric.
Unified approach to Laplace and Steklov eigenvalues via n-harmonic maps.
problem Eigenvalue problems on manifolds of arbitrary dimension.
method Unified description using n-harmonic maps. result Uncovering two new features of Steklov eigenvalues.
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.
Physics-Informed Neural Network (PINN) computes the Morse index of the critical catenoid.
problem Computing the Morse index of the critical catenoid
method Physics-Informed Neural Network (PINN) enforces parity and eigenvalue as trainable parameters
result Returns eigenvalues within 10−6 to 10−4 of exact values Optimizes metrics on surfaces for eigenvalues.
problem Finding optimal metrics for eigenvalues on surfaces.
method Combining constructions of Palais-Smale-like sequences and techniques from Karpukhin et al.
result Existence of optimal metrics for various eigenvalues on surfaces.
The i-th eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of fixed area. Extremal points of these functionals correspond to surfaces admitting minimal isometric immersions into spheres. Recently, critical metrics for the first eigenvalue were classified on tori a…
We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of t…
The paper finds a lower bound for a Neumann eigenvalue of surfaces with flat ends.
problem Finding a lower bound for the first Neumann eigenvalue of surfaces with asymptotically flat ends.
method Integration of Bouchner's formula with contributions from A. Lichnerowicz, S. Brendle, R. Tsiamis, and Poincare's constant.
result Obtains a lower bound for the first Neumann eigenvalue of properly embedded surfaces.
Gradient descent forces neural network eigenvalues to a specific threshold.
problem Understanding why gradient descent drives eigenvalues to a specific threshold.
method Introduced edge coupling, a functional on consecutive iterate pairs, to explain the trajectory towards the eigenvalue threshold.
result Gradient descent forces the Hessian eigenvalue to the threshold 2/η from arbitrary initialization. This thesis covers different aspects of the p-Laplace operators on Riemannian manifolds. Chapter 2. Potential theoretic aspects: the Khasmkinskii condition. Chapter 3: sharp eigenvalue estimates with Ricci curvature lower bounds. Chapter 4: Critical sets of (2-)harmonic functions.
Study on eigenvalue distribution of correlated time series, showing deformation of Marchenko-Pastur distribution.
problem Eigenvalue distribution of Wishart matrix with temporal correlation.
method Analysis of moments and convergence to deformed Marchenko-Pastur distribution for Gaussian process with temporal correlation.
result Eigenvalue distribution converges to deformed Marchenko-Pastur distribution with longer tail and higher peak.
The first eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of a given area. Critical points of this functional are called extremal metrics. The only known extremal metrics are a round sphere, a standard projective plane, a Clifford torus and an equilateral torus.…
High-dimensional models become unstable when sample size falls below a critical level, leading to a phase transition.
problem Instability in high-dimensional learning models when sample size is insufficient.
method Proved the necessity of a Fisher eigenvalue threshold for stability, introduced Fisher floor for verification.
result A sharp phase transition between reliable concentration and inevitable failure in high-dimensional learning.
This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some coni…
The use of certain critical-exponent Sobolev norms is an important feature of methods employed by Taubes to solve the anti-self-dual and similar non-linear elliptic partial differential equations. Indeed, the estimates one can obtain using these critical-exponent norms appear to be the best possible when one needs to b…
The paper explores how kernel eigenalignments affect generalization in KRR.
problem Achieving robust generalization in kernel methods.
method Direct connection between generalization and matrix eigenvectors/eigenvalues, focusing on finite-sample settings.
result Strong generalization requires increasing eigenvector alignment, eigenvalue magnitude, or gaps between eigenvalues.
Study eigenvalues of magnetic Steklov problem on Riemannian annuli.
problem Eigenvalues of magnetic Steklov problem on Riemannian annuli.
method Sharp upper bounds, maximizers, and existence of maximizers for eigenvalues.
result Existence of maximizers for the second normalized eigenvalue for rotationally invariant metrics.
Study new Willmore-type variational problem for foliated hypersurfaces.
problem New Willmore-type variational problem for hypersurfaces with foliations.
method Calculate first and second variations, find Euler-Lagrange equation, consider critical hypersurfaces.
result Found critical hypersurfaces of revolution as local minima for special variations.
We relate the distribution of eigenvalues of a random symmetric matrix in the Gaussian Orthogonal Ensemble to the distribution of critical values of a random linear combination of eigenfunctions of the Laplacian on a compact Riemann manifold. We then prove a central limit theorem describing what happens when the dimens…
We compare correlations and coherent structures in nuclei and financial markets. In the nuclear physics part we review giant resonances which can be interpreted as a coherent structure embedded in chaos. With similar methods we investigate the financial empirical correlation matrix of the DAX and Dow Jones. We will sho…
Study stability and bifurcation of liquid interfaces in cylindrical supports.
problem Stability and bifurcation of liquid interfaces in cylindrical support surfaces.
method Analysis of eigenvalues of the Jacobi operator, Plateau-Rayleigh instability, bifurcation theory.
result Conditions for the emergence of new morphologies and bifurcations from circular cylinders.
The paper studies critical points in overparameterized neural networks, identifying a star locus and degenerate critical points.
problem Understanding the geometry of loss functions in overparameterized neural networks.
method Identifying and analyzing components of the critical locus of the loss function L for overparameterized feedforward neural networks of depth ℓ≥4. result For very wide networks, all critical points are degenerate, and lower bounds on the number of zero eigenvalues of the Hessian are given.