Study eigenvalue variation in (p,q)-Laplacian on Ricci-harmonic flow.
problem Eigenvalue variation of (p,q)-Laplacian on evolving manifolds. method First variation formula for (p,q)-Laplacian eigenvalue on Ricci-harmonic flow. result Construct various monotonic quantities for (p,q)-Laplacian eigenvalue. Variational method for eigenvalues on manifolds.
problem Optimizing functionals involving eigenvalues of Riemannian manifolds.
method New Palais-Smale sequences and min-max methods for locally-Lipschitz functionals.
result Convergence of Palais-Smale sequences in Laplace and Steklov eigenvalues.
Study on eigenvalues of complex Hessian operator on pseudoconvex manifolds.
problem Eigenvalue problem for complex Hessian operator on pseudoconvex manifolds.
method Established C1,1-regularity and uniqueness of the first eigenfunction, derived variational formula for the first eigenvalue. result Derivation of a bifurcation-type theorem and geometric bounds for the eigenvalue.
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.
Study eigenvalues of conformal Laplacian under Sire-Xu normalization.
problem Existence and properties of extremal eigenvalues under a specific normalization.
method Variational analysis of eigenvalue functional under Sire-Xu normalization.
result Necessary conditions and existence results for extremal eigenvalues.
Eigenvalues of Steklov eigenproblems change predictably with boundary tweaks.
problem Understanding how Steklov eigenvalues respond to boundary changes.
method Analyzing smooth boundary perturbations of Steklov eigenvalues.
result Steklov eigenvalues are generically simple under such perturbations.
We study the existence and properties of metrics maximising the first Laplace eigenvalue among conformal metrics of unit volume on Riemannian surfaces. We describe a general approach to this problem and its higher eigenvalue versions via the direct method of calculus of variations. The principal results include the gen…
Paper confirms Yau's conjecture about sphere eigenvalues.
problem Yau's conjecture on eigenvalues of minimal hypersurfaces.
method Constructing a minimizing sequence in Sobolev space, using variational principle.
result First non-zero eigenvalue equals hypersurface dimension.
Derives a formula for the second variation of the Laplace eigenvalue functional on manifolds.
problem Calculating the second variation of the Laplace eigenvalue functional on closed manifolds.
method Derives a scale-invariant second variation formula for the Laplace eigenvalue functional.
result Proves that the canonical flat metric on a torus is not a maximal point of the functional in its conformal class.
Improved neural network inference with eigenvalue correction.
problem Inference of flexible variational posteriors is computationally expensive.
method Eigenvalue correction to matrix-variate Gaussian posterior.
result Empirically, the method outperforms existing algorithms.
Study on new Monge-Ampère functionals and their variational problems.
problem Existence and uniqueness of solutions for nonlinear eigenvalue problems.
method Introduction of a family of real Monge-Ampère functionals and proving Sobolev type inequalities.
result Existence of solutions for a nonlinear eigenvalue problem.
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.
Variational methods yield formulas for eigenvalues of elliptic operators, with applications to metric evolution.
problem Deriving formulas for eigenvalues of elliptic operators on compact manifolds.
method Variational methods applied to elliptic operators on compact Riemannian manifolds.
result Generic subsets of metrics yield simple spectra of elliptic operators.
Study finds lower bounds for energy on fibred manifolds using fiberwise symmetrization.
problem Finding lower bounds for energy functionals on fibred manifolds.
method Established a framework for fiberwise symmetrization to find lower bounds.
result Proved a comparison theorem for the first eigenvalue of the Laplacian on warped product manifolds.
New biharmonic Steklov problem on forms yields eigenvalue estimates.
problem Eigenvalue estimates for differential forms with curvature quantities.
method Introduced a new biharmonic Steklov problem and proved existence of a discrete spectrum.
result Established Kuttler-Sigillito inequalities connecting eigenvalues of differential forms.
We derive, under a technical assumption, the first variation formula for the eigenvalues of the Laplacian on a closed manifold evolving by the Ricci flow and give some applications.
We prove that in Riemannian manifolds the k-th Steklov eigenvalue on a domain and the square root of the k-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions. As an application, we derive a Weyl-type upp…
Sharp inequalities and eigenvalue problems on Finsler manifolds with nonnegative Ricci curvature.
problem Establishing sharp Morrey-Sobolev inequalities and eigenvalue problems on Finsler manifolds.
method Combining sharp isoperimetric inequality and anisotropic symmetrization argument.
result Existence and multiplicity of solutions for eigenvalue problems and elliptic PDEs.
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. Paper finds principal eigenvalue for infinity Laplacian in metric spaces.
problem Finding the principal eigenvalue of the infinity Laplacian in metric spaces.
method Direct PDE approach and Perron's method to establish existence of solutions.
result Existence of solutions to the infinity eigenvalue problem in metric spaces.
Researchers investigate extremal eigenvalues of GJMS operators in fixed conformal classes.
problem Investigating extremal eigenvalues of GJMS operators in fixed conformal classes.
method Developed a general framework for existence theory of extremals, defined and investigated generalised eigenvalues, and established semi-continuity results and Euler-Lagrange equations.
result Proved several new (non)-existence results for extremals of renormalised eigenvalues over the conformal class.
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 study finds a metric that maximizes the second eigenvalue of the Conformal Laplacian.
problem Maximizing the second eigenvalue of the Conformal Laplacian over conformal metrics.
method Analyzes properties of the Conformal Laplacian and constructs metrics to maximize eigenvalues.
result Existence of a metric that maximizes the second eigenvalue of the Conformal Laplacian.
Study on biharmonic Steklov problems with Neumann boundary conditions and eigenvalue estimates.
problem Biharmonic Steklov problems with Neumann boundary conditions.
method Introduced a biharmonic Steklov problem and proved its well-posedness. Established eigenvalue estimates using Kuttler-Sigillito inequalities.
result Eigenvalue estimates for the biharmonic Steklov problem with Neumann boundary conditions.
The paper compares Dirichlet and Neumann eigenvalues on various curved surfaces.
problem Comparing eigenvalues on curved surfaces.
method Variational principle of the Hodge Laplacian on 1-forms.
result Strict inequalities between Dirichlet and Neumann eigenvalues on specific surfaces.
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.
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.
In this paper, the author discusses the eigenvalues and entropies under the harmonic-Ricci flow, which is the Ricci flow coupled with the harmonic map flow. We give an alternative proof of results for compact steady and expanding harmonic-Ricci breathers. In the second part, we derive some monotonicity formulas for eig…
Optimizes functions on Lie groups using generalized eigenvalue problems.
problem Optimization on Lie groups with specific applications to eigenvalue problems.
method Generalizes NAG principle to Lie groups, resulting in continuous Lie-NAG dynamics converging to local optima.
result Discretized Lie-NAG dynamics yield structure-preserving optimization algorithms with faithful energy behavior.
Study on quasilinear elliptic equation on graphs with indefinite weights.
problem Existence and properties of solutions to quasilinear elliptic equations on graphs with indefinite weights.
method Variational methods and analysis of eigenvalue problems.
result Existence and monotonicity of the principal eigenvalue and positive solutions.
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 consider an analytic family of Riemannian metrics on a compact smooth manifold M. We assume the Dirichlet boundary condition for the η-Laplacian and obtain Hadamard type variation formulas for analytic curves of eigenfunctions and eigenvalues. As an application, we show that for a subset of all Cr Riemannian …
We compare the eigenvalues of the Dirac and Laplace operator on a two-dimensional torus with respect to the trivial spin structure. In particular, we compute their variation up to order 4 upon deformation of the flat metric, study the corresponding Hamiltonian and discuss several families of examples.
Ten sharp lower estimates of the first non-trivial eigenvalue of Laplacian on compact Riemannian manifolds are reviewed and compared. An improved variational formula, a general common estimate, and a new sharp one are added. The best lower estimates are now updated. The new estimates provide a global picture of what on…
For the dual operator sg′∗ of the linearization sg′ of the scalar curvature function, it is well-known that if kersg′∗=0, then sg is a non-negative constant. In particular, if the Ricci curvature is not flat, then sg/(n−1) is an eigenvalue of the Laplacian of the metric g. In this work, some…
The paper examines how the first Steklov-Dirichlet eigenvalue changes with the distance between two concentric circles.
problem Investigating the monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli.
method The approach involves showing differentiability, deriving integral expressions for the derivative, and using variational formulations to find upper and lower bounds.
result The paper proves the monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli with respect to the distance between the centers of the inner and outer boundaries.
We study the (massless) Dirac operator on a 3-sphere equipped with Riemannian metric. For the standard metric the spectrum is known. In particular, the eigenvalues closest to zero are the two double eigenvalues +3/2 and -3/2. Our aim is to analyse the behaviour of eigenvalues when the metric is perturbed in an arbitrar…
We derive the first and second variation formula for the Green's function pole's value of Paneitz operator on the standard three sphere. In particular it is shown that the first variation vanishes and the second variation is nonpositively definite. Moreover, the second variation vanishes only at the direction of confor…
The study examines metrics that extremize eigenvalues of a specific map on manifolds with boundary.
problem Variational properties of the spectrum of the Dirichlet-to-Robin map on manifolds with boundary.
method Analysis of the extremal metrics for the first and second normalized eigenvalues of the Dirichlet-to-Robin map.
result Existence and characterization of extremal metrics for the first and second eigenvalues of the Dirichlet-to-Robin map.
The paper studies how the first eigenvalue of a weighted p-Laplacian changes over time on Riemannian manifolds.
problem Evolution of the first eigenvalue of weighted p-Laplacian.
method Investigates monotonicity of the first eigenvalue problem along the Ricci-Bourguignon flow.
result First variation formula for eigenvalues and various monotonic quantities are derived.
We establish the existence of analytic curves of eigenvalues for the Laplace-Neumann operator through an analytic variation of the metric of a compact Riemannian manifold M with boundary by means of a new approach rather than Kato's method for unbounded operators. We obtain an expression for the derivative of the cur…
New method generalizes eigenvalue inequality to surfaces with boundaries.
problem Eigenvalue inequality for surfaces with boundaries.
method Generalized Rohleder's approach to differential forms, presenting Hodge-Laplacian spectrum.
result Obtained inequality for eigenvalues of Hodge-Laplacian and Dirichlet problems.
Study on null-torsion holomorphic curves in 6-sphere, focusing on their second variation.
problem Characterize the second variation of area for null-torsion holomorphic curves in the round 6-sphere.
method Analyzing the spectrum of the Jacobi operator for compact null-torsion holomorphic curves.
result For g≤6, the multiplicity of the lowest eigenvalue λ1=−2 is exactly 4d. EigenVI uses orthogonal function expansions for efficient variational inference.
problem Efficiently approximate complex distributions in variational inference.
method EigenVI constructs variational approximations using orthogonal function expansions, minimizing Fisher divergence.
result EigenVI provides more accurate approximations than existing methods for Gaussian BBVI.
In this paper we provide a detailed proof of the second variation formula, essentially due to Richard Hamilton, Tom Ilmanen and the first author, for Perelman's ν-entropy. In particular, we correct an error in the stability operator stated in Theorem 6.3 of [2]. Moreover, we obtain a necessary condition for linearly …
New algorithm samples from Ising models efficiently, even with outliers.
problem Sampling from Ising models with general interaction matrices.
method Combines MCMC and variational inference techniques.
result First polynomial time sampling algorithms for low-rank Ising models.
New proof for global rigidity of vertex scaling on polyhedral surfaces.
problem Global rigidity of vertex scaling on polyhedral surfaces.
method Elementary variational proof based on continuity of eigenvalues and extension of convex functions.
result Global rigidity of vertex scaling proved without involving 3D hyperbolic geometry.
The paper studies eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.
problem Eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.
method Extending the LII operator to Lν, establishing a general formula for eigenvalues, and applying it to estimate eigenvalues on Riemannian manifolds. result Established eigenvalue inequalities for the Lν2 operator on translating solitons and other geometric settings.