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. New estimate for stability eigenvalues of singular minimal hypersurfaces in spheres.
problem Estimating the first stability eigenvalue of singular minimal hypersurfaces in spheres.
method Extending an estimate by J. Simons to the singular setting.
result Any singular minimal hypersurface in Sn+1 has a first stability eigenvalue at most -2n. Study on stability of quaternion-Kähler manifolds using eigenvalue estimates.
problem Stability problem for positive quaternion-Kähler manifolds.
method Description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and symmetric 2-tensors. Improved eigenvalue estimates for the Hodge-Laplacian on 2-forms.
result Sharp lower bound for the first non-zero eigenvalue on the parallel subbundle Sym^2 E of the 2-form bundle.
The paper studies stability of domains for the first eigenvalue on Riemannian manifolds.
problem Stability of extremal domains for the first eigenvalue of the Laplacian operator.
method Second variation of the first Dirichlet eigenvalue, stability criterion, classification of stable domains.
result Classification of stable extremal domains in spheres and topological bounds for general compact surfaces.
Estimates eigenvalues on weighted manifolds with curvature.
problem Estimating eigenvalues of Dirichlet and Neumann problems.
method Using Bakry-Émery Ricci curvature.
result Established a stability condition for h-minimal hypersurfaces.
The paper compares two spectrum definitions and finds stability in one modification.
problem Generalizing eigenvalues to arbitrary functionals with stability.
method Comparison of Gromov's homotopy significant spectrum and Krasnoskii spectrum, with a modified definition of the homotopy significant spectrum.
result The modified homotopy significant spectrum is stable, and Cheeger constant corresponds to Krasnoskii eigenvalue.
We discuss stability of the first eigenvalue of the 1-Laplacian under perturbations of the domain.
Study on stability of Einstein metrics on symmetric spaces.
problem Stability of Einstein-Hilbert functional on compact symmetric spaces.
method Classification of irreducible representations and use of Casimir eigenvalues.
result Proves stability of Einstein metrics on quaternionic and Cayley projective plane, instability on other quaternionic Grassmannians.
The paper studies eigenvalues and stability of hypersurfaces in spheres.
problem Finding eigenvalues and stability of hypersurfaces in spheres.
method Derives equations for mean curvature and uses numerical methods to compute eigenvalues.
result Numerical computation of eigenvalues and stability indices for specific hypersurfaces.
Study on spectral stability of Riemannian coverings.
problem Stability of eigenvalues in Riemannian coverings.
method Analysis of Laplacian eigenvalues under finite coverings.
result Necessary conditions for spectral stability or instability.
Study on spectral stability of an embedded annulus under curve shortening and Ricci flows.
problem Spectral stability of Dirichlet eigenvalues on an evolving annulus.
method Variational formulas, Rellich-type identities, and harmonic capacity methods.
result Established quantitative bounds comparing the spectrum of the evolving annulus with a flat cylinder.
The paper refines the stability index for a specific minimal hypersurface and verifies Yau's conjecture.
problem Stability of minimal hypersurfaces in spheres and eigenvalue multiplicity.
method Analytical and numerical methods to study eigenvalues and stability indices.
result The multiplicity of the eigenvalue for the Carlotto-Schulz minimal embedding is at least 2n+1+n^2.
We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature orientable surfaces immersed in a Riemannian Killing submersion. As a consequence, the strong stability of such surfaces is studied. We also characterize constant mean curvature Hopf tori as the only ones att…
The paper studies stability and instability of minimal submanifolds in complex Einstein spaces.
problem Stability and instability of minimal submanifolds in complex Einstein spaces.
method Computation of index and nullity, investigation of stability, and algorithm for higher eigenvalues.
result Criterion for instability of minimal submanifolds in some cases.
Stability inequalities for specific solutions in high dimensions.
problem Stability of solutions to the one-phase Bernoulli problem.
method Proving strict stability inequalities for cohomogeneity one solutions with bi-orthogonal symmetry.
result Strict stability for cohomogeneity one solutions in dimensions 7 and above.
We prove stability results associated with upper bounds for the first eigenvalue of certain second order differential operators of divergence-type on hypersurfaces of the Euclidean space. We deduce some applications to r-stability as well as to almost-Einstein hypersurfaces.
New stability conditions for ZO methods reveal unique regularization effects.
problem Understanding optimization dynamics of ZO methods in deep learning.
method Explicit step size conditions and stability bounds derived for ZO methods.
result ZO methods operate near the edge of stability, with regularization effects specific to Hessian trace vs. eigenvalue.
The paper studies rigidity and stability of minimal submanifolds in hyperbolic space.
problem Conditions for a minimal submanifold to be totally geodesic.
method Analyzes the length of the second fundamental form and eigenvalues of the super stability operator.
result Sharp upper bounds for the first eigenvalue of the super stability operator for surfaces in hyperbolic 4-space.
Study on stability of constant mean curvature hypersurfaces in Riemannian manifolds.
problem Stability of constant higher mean curvature hypersurfaces in Riemannian manifolds.
method Introduced a new notion of stability and used two stability operators to relate it to the first eigenvalues. Applied to Space Forms and proved non-stability for certain hypersurfaces.
result Embedded rotational spheres with constant k-mean curvature in HnxR or SnxR are not stable.
Study shows no new eigenvalues in specific finite coverings.
problem Proving the absence of new eigenvalues in finite coverings.
method Analyzing spectral stability of finite coverings with specific conditions on Ricci curvature and representation theory.
result Non-existence of new eigenvalues in a specific range.
The paper finds new eigenfunctions for minimal immersions and their stability index.
problem Finding new eigenfunctions for minimal immersions and their stability index.
method Explicitly showed new eigenfunctions for the stability operator.
result The stability index of minimal immersions is at least kℓ+3k+3ℓ+8. Investment diversification affects financial stability, depending on network connectivity.
problem Analyzing stability of financial networks with diversified portfolios.
method Random matrix dynamical model with portfolio rebalancing, considering heterogeneity and diversification effects.
result Stability/instability transition depends on the largest eigenvalue of the random matrix.
The paper analyzes stability and generalization of GCNNs on graphs.
problem Theoretical understanding of GCNN models' generalization properties.
method Analysis of stability and derivation of generalization guarantees for semi-supervised graph learning.
result The stability of GCNN models depends on the largest absolute eigenvalue of the graph convolution filter.
We find out upper bounds for the first eigenvalue of the stability operator for compact constant mean curvature surfaces immersed into certain 3-dimensional Riemannian spaces, in particular into homogeneous 3-manifolds. As an application we derive some consequences for strongly stable surfaces in such ambient spaces. M…
Sharp upper bounds found for Steklov eigenvalues of a specific hypersurface.
problem Finding upper bounds for Steklov eigenvalues of a specific type of hypersurface.
method Analytical approach to compute upper bounds and prove stability properties.
result Sharp upper bounds Bn(L) and Bn for Steklov eigenvalues are derived. 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. Study eigenvalues of CROSS spaces with various metrics.
problem Eigenvalue of Laplace-Beltrami operator on CROSSes.
method Explicit formulae for the first eigenvalue of CROSSes with homogeneous metrics.
result Homogeneous metrics on CROSSes are isospectral if and only if they are isometric.
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.
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.
Adapts Stein's method for geometric inequalities, addressing boundary terms.
problem Geometric inequalities and their stability under constraints.
method Uses elliptic PDE with oblique boundary condition to handle boundary terms.
result Stability results for various geometric inequalities with respect to a new distance.
Develops a monitoring procedure to detect changes in large approximate factor models.
problem Detecting structural changes in large approximate factor models.
method Randomises the test statistic to create a sequence of i.i.d. statistics for monitoring changes.
result Very small probability of false detections and tight detection times of change-points.
We prove the stability of the Gieseker point of an irreducible homogeneous bundle over a rational homogeneous space. As an application we get a sharp upper estimate for the first eigenvalue of the Laplacian of an arbitrary Kaehler metric on a compact Hermitian symmetric spaces of ABCD--type.
Study on eigenvalues of Laplacian on anti-de Sitter manifolds.
problem Distribution and stability of eigenvalues in pseudo-Riemannian geometry.
method Analysis of L2-eigenvalues on anti-de Sitter manifolds. result Countably many L2-eigenvalues are stable under small deformations. New estimates show all stable Einstein manifolds are linear stable with respect to Perelman's ν-entropy.
problem Estimating the smallest eigenvalue of Laplace-Beltrami operator for stable Einstein manifolds.
method Estimating the smallest positive eigenvalue λ1 of the Laplace-Beltrami operator for standard Einstein manifolds (G/H,gst) and proving λ1>2E for all but 7 exceptions. result All stable Einstein manifolds found by Schwahn are linear stable with respect to Perelman's ν-entropy.
Stable solution found for manifold topology from boundary data.
problem Determining manifold properties from boundary data and eigenvalues.
method Quantitative stability estimates and unique continuation for the wave operator.
result Eigenvalues and boundary values determine a metric space close to the manifold.
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.
Estimates eigenvalues for surfaces in RP^3, proving a strict inequality.
problem Quantitative estimate of eigenvalues for surfaces in RP^3.
method Combination of canonical Veronese embedding, conformal test function method, rigidity argument, and classification of surfaces.
result Strict inequality for eigenvalues when χ(Σ) ≤ 0.
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.
Given a closed Riemannian manifold of dimenion less than eight, we prove a compactness result for the space of closed, embedded minimal hypersurfaces satisfying a volume bound and a uniform lower bound on the first eigenvalue of the stability operator. When the latter assumption is replaced by a uniform lower bound on …
Recent developments link Steklov eigenvalues to manifold geometry.
problem Steklov eigenvalues and eigenfunctions on compact Riemannian manifolds.
method Analytical and geometric approaches, including isoperimetric bounds, stability analysis, optimisation, and discretization.
result Connections between Steklov eigenvalues and manifold geometry, including optimisation and isospectrality.
In this paper, we study the Lagrangian F-stability and Hamiltonian F-stability of Lagrangian self-shrinkers. We prove a characterization theorem for the Hamiltonian F-stability of n-dimensional complete Lagrangian self-shrinkers without boundary, with polynomial volume growth and with the second fundamental form sati…
In this article, we prove new stability results for almost-Einstein hypersurfaces of the Euclidean space, based on previous eigenvalue pinching results. Then, we deduce some comparable results for almost umbilical hypersurfaces.
Let Σ be a compact immersed surface with constant weighted mean curvature Hf in a weighted manifold (M3,g,f). In this paper we obtain upper bounds for the first eigenvalue of the weighted Jacobi operator on Σ in terms of Hf and the curvature of the ambient. As consequence we obtain that there is no stable …
Stable algebraic filters improve neural network performance.
problem Improving neural network stability to deformations.
method Analyzed stability of algebraic filters and neural networks under deformations of the homomorphism.
result Stable algebraic filters have frequency responses whose derivative is inversely proportional to frequency.
Gradient descent performs well on weakly convex losses, offering generalization guarantees.
problem Learning with weakly convex losses using gradient descent.
method Analyzing the stability of gradient descent through the smallest eigenvalue of the Hessian.
result Generalization error bounds hold under a wider range of step sizes.
The paper proves geodesic balls maximize the first Steklov eigenvalue in non-compact symmetric spaces.
problem Characterizing geodesic balls in non-compact rank one symmetric spaces.
method Utilizing a weighted isoperimetric inequality on harmonic manifolds, the paper proves the maximization of the first Steklov eigenvalue.
result Geodesic balls uniquely maximize the first Steklov eigenvalue among domains of fixed volume in non-compact rank one symmetric spaces.
The paper computes spectra of Laplacian and Jacobi operators on rotational cmc hypersurfaces of spheres.
problem Computing spectra of Laplacian and Jacobi operators on rotational cmc hypersurfaces of spheres.
method Analyzing eigenvalues of second order Hill's equations and proving inequalities for stability index and eigenvalues.
result Proves that the stability index of minimal rotational examples is greater than 3n+4 and there are at least 2 positive Laplacian eigenvalues smaller than n. Paper examines stability of quadratic curvature functionals on product Einstein manifolds.
problem Understanding stability of critical points of quadratic curvature functionals on product Einstein manifolds.
method Analyzes Riemannian functionals defined by L2-norms of Ricci, scalar, Weyl, and Riemannian curvatures. result Product of a spherical space form and a compact hyperbolic manifold is unstable for some quadratic functionals if the first eigenvalue of the Laplacian of the hyperbolic manifold is sufficiently small.