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.
Proves existence of maximizers for eigenvalue optimization on manifolds.
problem Eigenvalue optimization on Riemannian manifolds of dimension m≥3. method Use of topological tensor products to analyze eigenvalue functionals.
result Absolutely continuous maximizers are induced by p-harmonic maps into spheres. Optimal lower bounds for eigenvalues of Dirac-Witten operator on certain submanifolds.
problem Estimating eigenvalues of the Dirac-Witten operator on specific submanifolds.
method Optimal lower bounds derived using intrinsic and extrinsic expressions.
result Limiting-cases of eigenvalues studied and optimal bounds obtained.
Optimizes eigenvalue bounds for submanifold Dirac operators.
problem Estimating eigenvalues of submanifold Dirac operators.
method Optimal lower bounds derived using intrinsic and extrinsic expressions.
result Optimal eigenvalue bounds established for submanifold Dirac operators.
This paper is a tutorial for eigenvalue and generalized eigenvalue problems. We first introduce eigenvalue problem, eigen-decomposition (spectral decomposition), and generalized eigenvalue problem. Then, we mention the optimization problems which yield to the eigenvalue and generalized eigenvalue problems. We also prov…
Optimal constants for isoperimetric inequalities involving Steklov eigenvalues on surfaces are determined.
problem Determining optimal constants for isoperimetric inequalities involving Steklov eigenvalues on surfaces.
method Analyzing Riemannian surfaces with boundary, considering both given topology and conformal class, and proving inequalities relating conformal invariants and eigenvalues.
result New examples of topological disks realizing optimal constants and inequalities relating conformal invariants of Steklov eigenvalues on surfaces and disks are provided.
Study on shape optimization for specific eigenvalue problems on domains.
problem Shape optimization of eigenvalue problems for fourth order Steklov.
method Asymptotic expansion and sharp upper bound derivation.
result Derivation of eigenvalue spectra and shape optimization conclusions.
Optimizes eigenvalues on surfaces with symmetries.
problem Maximizing Laplace and Steklov eigenvalues on Riemann surfaces with symmetries.
method Simplifies existing techniques for conformal class optimization.
result Proves existence and regularity of maximizers for Laplace and Steklov eigenvalues.
Optimizes the first eigenvalues of Riemann surfaces for large genus.
problem Finding optimal lower bounds for first eigenvalues of Riemann surfaces.
method Analyzing shortest multi-closed curves to establish a new lower bound.
result The first eigenvalue of a Riemann surface is greater than a specific formula involving the genus and a constant.
Optimizes metrics for the first curl eigenvalue on 3-manifolds.
problem Finding optimal metrics for minimizing the first curl eigenvalue.
method Analyzes metrics that minimize the first curl eigenvalue among metrics of the same volume in the same conformal class.
result Proves that S3 and RP3 are local minimizers for the first curl eigenvalue. Optimizes maps and eigenvalues on manifolds.
problem Maximizing the first eigenvalue of a manifold's Laplacian.
method Formulates dual optimization problems involving maps and eigenvalues.
result Proves a Nadirashvili-type theorem for eigenvalue maximization.
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.
In this paper, we obtain "universal" inequalities for eigenvalues of the weighted Hodge Laplacian on a compact self-shrinker of Euclidean space. These inequalities generalize the Yang-type and Levitin-Parnovski inequalities for eigenvalues of the Laplacian and Laplacian. From the recursion formula of Cheng and Yang \ci…
Numerical methods solve Steklov eigenvalue problems to generate free boundary minimal surfaces.
problem Generating free boundary minimal surfaces using Steklov eigenvalue problems.
method Maximizing Steklov eigenvalues over a class of metrics, using conformal uniformization and gradient-based optimization.
result Numerical solutions for free boundary minimal surfaces with various boundary components.
Study bounds Neumann and Steklov eigenvalues on manifolds and submanifolds.
problem Bounding Neumann and Steklov eigenvalues on manifolds and submanifolds.
method Using conformal and extrinsic volumes, the paper derives upper bounds for eigenvalues.
result Upper bounds for harmonic mean of Neumann and Steklov eigenvalues.
The paper finds the largest eigenvalue for a specific type of domain in hyperbolic space.
problem Finding the domain with the largest first eigenvalue for a given volume and boundary conditions.
method Shape optimization for the first eigenvalue of the p-Laplace operator in hyperbolic space.
result The concentric annular region maximizes the first eigenvalue among multiply-connected domains.
Study on Kähler manifolds shows rigidity of eigenvalues with positive Ricci bound.
problem Optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound.
method Established optimal rigidity results for eigenvalues on Kähler manifolds with positive Ricci lower bound.
result Complex projective space is the only Kähler manifold with the largest multiplicity of the first eigenvalue.
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.
Study eigenvalues and shapes, proving sharp inequalities for Steklov eigenvalues.
problem Eigenvalue continuity and shape optimization for Laplace and Steklov problems.
method Variational eigenvalue analysis, Sobolev space convergence, shape optimization techniques.
result Sharp isoperimetric inequalities for Steklov eigenvalues, upper bound 8πk for k-th perimeter-normalized eigenvalue. We consider a shape optimization problem for the first mixed Steklov-Dirichlet eigenvalues of domains bounded by two balls in two-point homogeneous space. We give a geometric proof which is motivated by Newton's shell theorem
Paper proves Faber-Krahn inequalities for weighted Laplacian eigenvalues.
problem Proving inequalities for eigenvalues of weighted Laplacian.
method Analyzing Robin boundary conditions on Rn and Hn. result Optimal domain for eigenvalues is a ball centered at the origin.
This paper deals with eigenvalue optimization problems for a family of natural Schrödinger operators arising in some geometrical or physical contexts. These operators, whose potentials are quadratic in curvature, are considered on closed surfaces immersed in space forms and we look for geometries that maximize the eige…
In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of L operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau \cite{SY}. By the eigenvalue estimates, we can prove a compac…
Higher surgeries preserve Steklov spectra in 3D and above.
problem Effect of topology changes on Steklov eigenvalues in higher dimensions.
method Perform surgeries of codimension 2 or higher on compact manifolds.
result Topology changes do not affect Steklov spectra in dimensions 3 and above.
Study eigenvalues and eigenfunctions of fourth-order operators in annuli, proving optimal estimates and non-radiality.
problem Eigenvalue and eigenfunction analysis of fourth-order operators in degenerating annuli.
method Optimal estimates and non-radiality results for eigenfunctions in annuli.
result Nigh optimal estimate for the first eigenvalue and non-radiality of eigenfunctions in degenerating annuli.
Paper proposes a new optimization framework for learning eigenfunctions of operators.
problem Computing eigenvalue decomposition of high-dimensional operators.
method Operator SVD with Neural Networks via Nested Low-Rank Approximation.
result Proposed method efficiently learns top-L singular values and functions in the correct order.
Study shows optimal spectral gaps diminish in large genus surfaces.
problem Optimizing spectral gaps in large genus surfaces.
method Analysis of Weil-Petersson probability and eigenvalues of Laplacian.
result Probability of optimal spectral gaps vanishes as genus increases.
Study rigidity of minimal Legendrian submanifolds in spheres via eigenvalues.
problem Rigidity of minimal Legendrian submanifolds in unit Euclidean spheres.
method Using Lu's inequality and eigenvalues of fundamental matrices to establish pinching theorems.
result Optimal pinching theorem and rigidity theorem for submanifolds of all dimensions.
We propose a method to clean covariance matrices of nonstationary systems by using time-independent eigenvalues.
problem Noise in covariance matrices of nonstationary systems with time-independent eigenvalues.
method Data-driven approach to use independent eigenvalues encoding long-term influence of future on present.
result Our method outperforms optimal stationary methods for filtering covariance matrix and its inverse.
For n≥7, we give the optimal estimate for the second eigenvalue of Paneitz operators for compact n-dimensional submanifolds in an (n+p)-dimensional space form.
Optimizes Dirac eigenvalue bound using curvature and quasi-spherical metrics.
problem Optimizing the first Dirac eigenvalue of hypersurfaces.
method Combining positive mass theorem and quasi-spherical metrics.
result Proves optimal upper bound for first Dirac eigenvalue.
The paper improves eigenvalue estimates for manifolds with Ricci curvature conditions.
problem Eigenvalue estimates for manifolds with Ricci curvature conditions.
method Proves eigenvalue estimates using a Kato condition on the negative part of Ricci curvature.
result Optimal eigenvalue estimates for Zhong-Yang type and Cheng-type bounds.
We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet L…
Sharp eigenvalue bounds on metric measure spaces extend Cheng's theorem.
problem Extending Cheng's eigenvalue comparison theorem to non-smooth spaces.
method Localization technique, synthetic Ricci curvature bounds via optimal transport.
result Sharp upper bounds on eigenvalues in metric measure spaces.
We relate small 1-form Laplacian eigenvalues to relative cycle complexity on closed hyperbolic manifolds: small eigenvalues correspond to closed geodesics no multiple of which bounds a surface of small genus. We describe potential applications of this equivalence principle toward proving optimal torsion homology growth…
We consider the lower order eigenvalues of poly-Laplacian with any order on spherical domains. We obtain universal inequalities for them and show that our results are optimal.
Existence of harmonic maps from higher-dimensional manifolds to spheres proven.
problem Proving existence of nonconstant harmonic maps from arbitrary manifolds to spheres.
method Using optimal regularity and eigenvalue optimization on manifolds.
result First general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets.
On a closed weighted Riemannian manifold with nonnegative Bakry-Émery Ricci curvature, it is shown that the ratio of the k-th to first eigenvalues of the weighted Laplacian is dominated by 641k2, using an argument via the Cheeger constant. While improving the previous exponential upper bound, the order of k here…
The paper proves stability of eigenvalue inequalities on surfaces.
problem Stability of isoperimetric inequalities for Laplace eigenvalues on surfaces.
method Employing eigenvalues of measures and Sobolev space W−1,2, the paper proves stability estimates for the first and second nonzero Laplace eigenvalues on surfaces. result Metrics almost maximizing the normalized eigenvalue are W−1,2-close to a maximal metric. In this paper, we study the first eigenvalue of Jacobi operator on an n-dimensional non-totally umbilical compact hypersurface with constant mean curvature H in the unit sphere Sn+1(1). We give an optimal upper bound for the first eigenvalue of Jacobi operator, which only depends on the mean curvature H and …
Upper bounds for Steklov eigenvalues of warped products are derived.
problem Finding upper limits for Steklov eigenvalues of warped product manifolds.
method Using volume, boundary volume, fiber Laplace eigenvalues, and warping function norms.
result Optimal upper bounds and stability estimates for eigenvalues are obtained.
In this paper, we consider the eigenvalue problem for Hodge-Laplacian on a Riemannian manifold M isometrically immersed into another Riemannian manifold Mˉ for arbitrary codimension. We first assume the pull back Weitzenböck operator (defined in Section 2) of Mˉ bounded from below, and obtain an extrinsic…
Constructs minimal surfaces in balls, maximizing eigenvalues.
problem Finding minimal surfaces in Euclidean balls with controlled topology.
method Maximizing the first non-trivial Steklov eigenvalue for isoperimetric problems.
result Constructs free boundary minimal immersions with controlled topology.
Derives an inequality for submanifolds in spheres.
problem Characterize submanifolds in spheres.
method Derives an integral inequality.
result Characterizes spheres.
In this paper we study the eigenvalues of buckling problem on domains in a unit sphere. By introducing a new parameter and using Cauchy inequality, we optimize the inequality obtained by Wang and Xia in [12].
The paper analyzes the performance of delay-based reservoir computing using eigenvalue analysis.
problem Quantifying the performance of delay-based reservoir computing.
method Eigenvalue analysis of the dynamical system to predict reservoir computing performance.
result The performance of a reservoir computing system can be predicted by analyzing the small signal response and eigenvalue spectrum.
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.
Improved portfolio optimization using Kendall-like correlation coefficients.
problem Accurate estimation of eigenvectors in data-poor regimes for portfolio optimization.
method Developed generalized correlation coefficients based on Kendall's rank correlation.
result Markowitz portfolios with lower out-of-sample risk using these coefficients.