Proves Payne conjecture for buckling and membrane eigenvalues.
problem Proving Payne conjecture for buckling and membrane eigenvalues.
method Analytical proof for buckling and membrane eigenvalues.
result Proves Payne conjecture for n-dimensional case (n≥2). Study on Neumann eigenvalues controlled by domain isoperimetric ratio.
problem Control the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue.
method Combination of analytical and numerical results, related to Yau's conjecture.
result Neumann eigenvalues are controlled by the isoperimetric ratio of the domain.
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.
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.
The eigenvalue conjecture is supported for colored Alexander polynomials.
problem Supporting the eigenvalue conjecture for colored Alexander polynomials.
method Connecting Alexander polynomials and eigenvalues of braid group generators.
result Support for the eigenvalue conjecture for i>2, where direct evaluation is difficult.
Improved lower bounds for poly-Laplacian eigenvalues in arbitrary dimensions.
problem Lower bounds for higher eigenvalues of the poly-Laplacian operator.
method Sharp inequalities and eigenvalue bounds in low and arbitrary dimensions.
result Improved lower bounds for eigenvalues of the poly-Laplacian in arbitrary dimensions.
Proves Pólya's conjecture for thin products and Riemannian manifolds.
problem Proving Pólya's conjecture for specific geometric domains.
method Analyzes thin products and Riemannian manifolds, proving inequalities for eigenvalues.
result Proves Pólya's conjecture for thin products and related Riemannian manifolds.
The paper proves Simon's conjecture for closed Einstein spaces, setting eigenvalue bounds.
problem Proving Simon's conjecture for closed Einstein spaces.
method Developed a method to prove Simon's conjecture for closed Einstein spaces.
result Proved Simon's conjecture for closed Einstein spaces, setting eigenvalue bounds.
Liu's paper contains an error regarding eigenvalues.
problem Eigenvalues of Dirichlet and buckling problems.
method Review and identification of an error.
result Error in the Payne conjecture for eigenvalues.
A well known conjecture of Yau states that the first eigenvalue of every closed minimal hypersurface Mn in the unit sphere Sn+1(1) is just its dimension n. The present paper shows that Yau conjecture is true for minimal isoparametric hypersurfaces. Moreover, the more fascinating result of this paper is that t…
Paper proves new inequalities for hyperbolic space Laplacian eigenvalues.
problem Universal inequalities for eigenvalues of the Dirichlet Laplacian.
method Proves new inequalities for eigenvalues on hyperbolic space.
result Verifies Cheng's conjecture up to a small loss.
The paper confirms Escobar's conjecture on Steklov eigenvalues.
problem The first nonzero Steklov eigenvalue of a manifold with specific curvature conditions.
method Combination of weighted Reilly type formula and Pohozaev type identity.
result The conjecture is confirmed for nonnegative sectional curvature.
Falsehood of Pólya's conjecture for spheres shown.
problem Disproving Pólya's eigenvalue conjecture for spheres.
method Comparison of Laplace spectrum and Weyl function of spheres.
result No analogue of Pólya's conjecture holds for spheres.
Three counterexamples show higher eigenvalue multiplicities than conjectured.
problem Determining the maximum eigenvalue multiplicity for closed hyperbolic surfaces.
method Applying the twisted Selberg trace formula to induced representations of triangle groups.
result Found counterexamples with higher eigenvalue multiplicities than previously conjectured.
In this note, we present some interesting observations on the Schiffer's conjecture, interior transmission eigenvalue problem and their connections to singular and nonsingular invisibility cloaking problems of acoustic waves.
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.
Upper bound found for Steklov eigenvalues counting function.
problem Counting Steklov eigenvalues on compact manifolds with boundary.
method Used Weyl's law and Pólya's Conjecture in the Steklov case.
result Obtained an upper bound for the counting function.
We give a new estimate on the lower bound of the first Dirichlet eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature and provide a solution for a conjecture of H. C. Yang.
The paper classifies biconservative Lorentz hypersurfaces with complex eigenvalues.
problem Classifying biconservative Lorentz hypersurfaces with complex eigenvalues.
method Analyzing biharmonic and biconservative submanifolds in E1n+1. result Every biconservative Lorentz hypersurface M1n in E1n+1 with complex eigenvalues has constant mean curvature. The study sets lower bounds for eigenvalue sums of Laplacian on bounded domains and spheres.
problem Establishing lower bounds for eigenvalue sums of the Laplacian.
method Extending known results on eigenvalues of Laplacian for bounded domains, spheres, and surfaces.
result Improved lower bounds for eigenvalue sums, connecting to conjectures and extending known results.
For a given bounded domain Ω⊂Rn with C1-smooth boundary, we prove the Pólya conjecture for the Neumann eigenvalues. In other words, we prove that \begin{eqnarray*} μ_{k+1}\le \frac{(2π)^2k^{2/n}}{(ω_n \cdot \mbox{vol}\, (Ω))^{2/n}} \quad \;\; \mbox{for all} \;\; k=0,1,2,3,\cdots,\end{eqnarray*} wher…
The paper proves that most metrics satisfy a strong version of Arnold's conjecture for Laplace eigenvalues.
problem Understanding metrics that satisfy a strong version of Arnold's conjecture for Laplace eigenvalues.
method Using geometric characterizations and perturbation theory, the paper proves the conjecture for most metrics.
result The Strong Arnold Hypothesis is satisfied for all metrics except for a set of infinite codimension.
In this paper, we investigate the Dirichlet problem of Laplacian on complete Riemannian manifolds. By constructing new trial functions, we obtain a sharp upper bound of the gap of the consecutive eigenvalues in the sense of the order, which affirmatively answers to a conjecture proposed by Chen-Zheng-Yang. In addition,…
Proven isoperimetric inequality for Witten-Laplacian eigenvalues.
problem Proving isoperimetric inequality for lower order nonzero Neumann eigenvalues of Witten-Laplacian.
method Analytical proof using Euclidean and hyperbolic spaces.
result Strengthens Szegő-Weinberger inequality and covers Xia-Wang's progress.
This is a continuation of Tang and Yan, which investigated the first eigenvalues of minimal isoparametric hypersurfaces with g=4 distinct principal curvatures and focal submanifolds in unit spheres. For the focal submanifolds with g=6, the present paper obtains estimates on all the eigenvalues, among others, giving…
The paper characterizes Pólya's conjecture for spheres and hemispheres, deriving inequalities and bounds.
problem Characterizing Pólya's conjecture for eigenvalues on spheres and hemispheres.
method Analyzing eigenvalues of the Laplace-Beltrami operator on spheres and hemispheres, deriving inequalities and bounds.
result Pólya's conjecture holds for hemispheres in the Neumann case but not in the Dirichlet case when n>2. Researchers decompose curvature to confirm Hopf conjecture and prove new rigidity theorems.
problem Confirming the Hopf conjecture on compact Riemannian manifolds of even dimension.
method Decomposing the curvature operator into Hermitian components and developing eigenvalue criteria for sectional curvature.
result Prove vanishing theorems for Betti numbers under integral bounds on the Weyl tensor and confirm the Hopf conjecture for manifolds with sufficiently small Weyl curvature.
Mathematicians decode geometric properties from eigenvalues over 112 years.
problem Recovering geometric properties from eigenvalues of Laplace equations.
method Analyzing the relationship between eigenvalues, domain volume, and dimensionality.
result Deep connection between eigenvalues and geometric properties elucidated by Weyl's law.
The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot; however, in full generality it is proven only for zeta functions associated to a polynomial in two variables. In this article we consider zeta…
We prove several results about the multiplicity of the first Steklov eigenvalues on compact surfaces with boundary. We improve some bounds on the multiplicity, especially for the first eigenvalue, and we prove they are sharp on some surfaces of small genus. In a previous article, we defined a new chromatic invariant of…
Paper proves a metric on a genus two surface maximizes Laplacian eigenvalue.
problem Maximizing the first eigenvalue of the Laplacian on a closed surface.
method Analyzes a specific singular metric on the Bolza surface.
result Proves a metric maximizes the Laplacian eigenvalue on a genus two surface.
In this paper, we investigate eigenvalues of the Dirichlet problem and the closed eigenvalue problem of drifting Laplacian on the complete metric measure spaces and establish the corresponding general formulas. By using those general formulas, we give some upper bounds of consecutive gap of the eigenvalues of the eigen…
In this paper, we derive the CR Reilly's formula and its applications to studying of the first eigenvalue estimate for CR Dirichlet eigenvalue problem and embedded p-minimal hypersurfaces. In particular, we obtain the first Dirichlet eigenvalue estimate in a compact pseudohermitian (2n+1)-manifold with boundary and the…
Study confirms conjecture for a specific type of Lie group metrics.
problem Establishing a bound for the smallest Laplace eigenvalue for naturally reductive metrics.
method Analyzing naturally reductive left-invariant metrics on compact simple Lie groups.
result The conjecture is confirmed for a specific subclass of Lie group metrics.
In this paper, we partially solve Yau' Conjecture of the first eigenvalue of an embedded compact minimal hypersurface of unit sphere Sn+1(1), i.e., Corollary 1.2. In particular, Corollary 1.3 proves that the condition ∫Ω1∣∇u∣2=(n+1)∫Ω1u2 is naturally true and meaningful in …
In this paper, we establish sharp inequalities for four kinds of classical eigenvalues on a bounded domain of a Riemannian manifold. We also establish asymptotic formulas for the eigenvalues of the buckling and clamped plate problems. In addition, we give a negative answer to the Payne conjecture for the one-dimensiona…
By the calculation of the gap of the consecutive eigenvalues of Sn with standard metric, using the Weyl's asymptotic formula, we know the order of the upper bound of this gap is kn1. We conjecture that this order is also right for general Dirichlet problem of the Laplace operator, which is optimal…
The paper proves diameter bounds and finiteness for amply regular graphs.
problem Proving diameter bounds and finiteness for amply regular graphs.
method Improved curvature estimates and new Bakry-Émery curvature estimates.
result There are only finitely many amply regular graphs with specific parameters.
Survey on isoparametric theory applications and results.
problem Yau's conjecture and problems in isoparametric theory.
method Various examples and counterexamples in isoparametric theory.
result Affirmative answer to Yau's conjecture on the first eigenvalue of Laplacian.
In this paper, we study lower bounds for higher eigenvalues of the Dirichlet eigenvalue problem of the Laplacian on a bounded domain Ω in Rn. It is well known that the k-th Dirichlet eigenvalue λk obeys the Weyl asymptotic formula, that is, \[ λ_k\sim\frac{4π^2}{(ω_n\mathrm{vol}Ω)^\frac{2}{n}}k^\frac…
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…
New families of non-tiling domains satisfy Pólya's conjecture.
problem Finding non-tiling domains that satisfy Pólya's conjecture.
method Analyzing partitioning and eigenvalue orders of domains.
result Existence of families of non-tiling domains satisfying Pólya's conjecture.
Study geometric properties of generalized vacuum static spaces.
problem Estimating geometric properties of generalized φ-vacuum static spaces. method Proving estimates for φ-scalar curvature and first eigenvalue of the Jacobi operator, and rigidity under various geometric assumptions. result Proved a result related to the Cosmic no-hair conjecture.
Lu's conjecture proven for minimal surfaces in codimension two.
problem Proving Lu's conjecture for minimal surfaces in codimension two.
method Analyzing eigenvalues of Lu's fundamental matrix and squared norm of the second fundamental form.
result Lu's second-gap conjecture holds for minimal surfaces in codimension two.
Study on Lawson surfaces' first Laplace eigenvalue using symmetry and algebraic methods.
problem Yau's conjecture on first eigenvalue of minimal hypersurfaces in the sphere.
method Symmetry-based approach exploiting discrete reflection symmetries and algebraic structure of reflection groups.
result Equality λ1(ξ_{m,k})=2 for Lawson surfaces with m and k even.
Paper proves surfaces with specific symmetries have the first Steklov eigenvalue.
problem Proving surfaces with certain symmetries have the first Steklov eigenvalue.
method Analyzing surfaces with reflection planes and genus zero.
result Surfaces with n distinct reflection planes have the first Steklov eigenvalue. Inspired by Katz-Mazur theorem on crystalline cohomology and by Eskin-Kontsevich-Zorich's numerical experiments, we conjecture that the polygon of Lyapunov spectrum lies above (or on) the Harder-Narasimhan polygon of the Hodge bundle over any Teichmüller curve. We also discuss the connections between the two polygons a…
The paper bounds eigenvalue multiplicities for hyperbolic surfaces using short geodesics.
problem Bounding the multiplicity of Laplacian eigenvalues for hyperbolic surfaces.
method Using the number of short closed geodesics and surface genus.
result Upper bounds on eigenvalue multiplicities, showing sublinear behavior under certain conditions.