arXiv research

New framework for studying eigenvalue functionals of metrics.

Paper derives second variation formula for eigenvalue functionals on surfaces.

The study classifies solutions to a specific eigenvalue problem and identifies the critical catenoid.

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.

Hot spots conjecture proven for small eigenvalue 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. $λ_1^\mathbb{T}$, 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.

Optimizing quantum graphs yields geodesic nets on surfaces.

New proofs of Cheeger-like inequalities for coexact 1-forms on hyperbolic manifolds.

Jakobson and Nadirashvili \cite{JN} constructed a sequence of eigenfunctions on $T^2$ 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 $Σ^2 \subset B^3$ which intersects $\partial B^3$ 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.

The paper finds rotationally symmetric critical metrics for Laplace eigenvalues on tori.

Study improves regularity estimates for harmonic maps into ellipsoids.

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 $(\mathcal{M},g)$. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue $μ_2(Ω)$ 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 $-\frac{1}{r}$, 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 $k\geq 3$. For $k=1$ the classical…

Study extends eigenvalue formulas to weighted manifolds and proves global rigidity theorems.

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.

Study eigenvalues of Dirac operator on surfaces, proving existence and deriving inequalities.

Proves existence of non-planar minimal disks in ellipsoids.

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.

Unified approach to Laplace and Steklov eigenvalues via $n$-harmonic maps.

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

Physics-Informed Neural Network (PINN) computes the Morse index of the critical catenoid.

Optimizes metrics on surfaces for eigenvalues.

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 determine all critical configurations for the Area function on polygons with vertices on a circle or an ellipse. For isolated critical points we compute their Morse index, resp index of the gradient vector field. We relate the computation at an isolated degenerate point to an eigenvalue question about combinations. …

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.

Gradient descent forces neural network eigenvalues to a specific threshold.

We characterize those complete K{ä}hler manifolds supporting a nonconstant real-valued function with critical points whose Hessian is complex linear, has pointwise two eigenvalues and whose gradient is a Hessian-eigenvector.

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.

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.

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.

Study eigenvalues of magnetic Steklov problem on Riemannian annuli.

Study new Willmore-type variational problem for foliated hypersurfaces.

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.

The paper studies critical points in overparameterized neural networks, identifying a star locus and degenerate critical points.