The paper explores inequalities between eigenvalues on Riemannian manifolds.
problem Investigating relationships between eigenvalues on Riemannian manifolds.
method Constructing gradient estimates for a first eigenfunction to derive inequalities.
result Obtained some relationships between weighted p-Laplacian first eigenvalues. Study eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
problem Finding lower bounds for eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
method Analytical proofs for both Neumann and Dirichlet boundary conditions.
result Established lower bounds for eigenvalues on compact quaternionic Kähler manifolds.
Sharp bounds derived for the first two Steklov eigenvalues of exterior domains.
problem Finding bounds for the first two eigenvalues of Steklov eigenvalue problems on exterior domains.
method Sharp lower and upper bounds derived using the support function and distance function to the origin of the boundary.
result Sharp bounds for the first two eigenvalues of Steklov eigenvalue problems on exterior domains.
Sharp lower bound for first Neumann eigenvalue found in terms of diameter and width.
problem Finding the minimum value of the first Neumann eigenvalue for convex domains.
method Proved the sharp lower bound using diameter and width.
result Sharp lower bound for the first Neumann eigenvalue established.
In this paper, two interesting eigenvalue comparison theorems for the first non-zero Steklov eigenvalue of the Laplacian have been established for manifolds with radial sectional curvature bounded from above. Besides, sharper bounds for the first non-zero eigenvalue of the Wentzell eigenvalue problem of the weighted La…
In this paper, we study the first two eigenvalues of the buckling problem on spherical domains. We obtain an estimate on the second eigenvalue in terms of the first eigenvalue, which improves one recent result obtained by Wang-Xia in [7].
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. The paper studies eigenvalue problems on manifolds and recovers known inequalities.
problem Eigenvalue problems on complete compact Riemannian manifolds with Dirichlet boundary conditions.
method Cheng comparison estimates, Faber-Krahn inequality, Cheeger estimates.
result Eigenvalue bounds and convergence to Cheeger's constant as p,qo1,1. Study eigenvalues of p-Laplacian on Kähler manifolds, proving lower bounds.
problem Eigenvalue problem for the p-Laplacian on Kähler manifolds.
method Lower bounds derived using dimension, diameter, curvature bounds.
result Sharp lower bounds for the first Dirichlet eigenvalue of the p-Laplacian.
Researchers prove rigidity of first conformal Steklov eigenvalue on specific shapes.
problem Rigidity of the first conformal Steklov eigenvalue on annuli and Möbius bands.
method Proof relies on uniqueness results, compactness theorem, and asymptotic control of Steklov eigenvalues.
result Rigidity of the first conformal Steklov eigenvalue on annuli and Möbius bands proved.
Geodesic disks maximize the first non-trivial Neumann eigenvalue on spheres.
problem Maximizing the first non-trivial Neumann eigenvalue on spheres.
method Proving maximizers are geodesic disks.
result Geodesic disks maximize the first non-trivial Neumann eigenvalue.
Sphere theorems for p-Laplacian eigenvalues established.
problem Sphere theorems for p-Laplacian eigenvalues.
method Established sphere theorems for p-Laplacian eigenvalues.
result Sphere theorems for p-Laplacian eigenvalues established.
The paper finds lower bounds for the first eigenvalue of p-Laplacian in specific manifolds.
problem Finding lower bounds for the first eigenvalue of p-Laplacian in Riemannian manifolds.
method Established and enhanced lower bounds for the eigenvalue under specific conditions.
result Provided an estimation for the first Dirichlet eigenvalue in asymptotically hyperbolic Einstein manifolds.
In study of eigenvalue problems, a classical problem is the Stekloff eigenvalue problem. There are many estimates of the first non- zero Stekloff eigenvalue, including a sharp estimate on surfaces, obtained by Escobar in "The geometry of the first non-zero Stekloff eigenvalue, J. Funct. Anal. 150 (1997)". In this paper…
In two previous papers, we started a study of the first eigenvalue of the Dirac operator on compact spin symmetric spaces, providing, for symmetric spaces of "inner" type, a formula giving this first eigenvalue in terms of the algebraic data of the groups involved. We conclude here that study by giving the explicit exp…
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…
Improved lower bound for the first eigenvalue of embedded minimal hypersurfaces in the unit sphere
problem First eigenvalue of embedded minimal hypersurfaces
method Establishing an improved lower bound
result Better than Duncan-Sire-Spruck's bound
Geodesic balls with non-negative Ricci curvature have a sharp lower bound on their first Dirichlet eigenvalue.
problem Finding a sharp lower bound for the first Dirichlet eigenvalue of geodesic balls.
method Quantitative explicit inequality linking the width of geodesic balls to the spectral gap.
result A quantitative inequality relating the width of geodesic balls to the spectral gap between the first Dirichlet eigenvalue and its lower bound.
Improved bound on first eigenvalue of minimal surfaces in S3.
problem Bounding the first eigenvalue of minimal surfaces in S3. method Proved λ1≥1+εg for embedded minimal surfaces Σ in S3. result Improved bound on the first eigenvalue of λ1. We study sharp asymptotics of the first eigenvalue on Riemannian surfaces obtained from a fixed Riemannian surface by attaching a collapsing flat handle or cross cap to it. Through a careful choice of parameters this construction can be used to strictly increase the first eigenvalue normalized by area if the initial su…
In this paper, we mainly investigate continuity, monotonicity and differentiability for the first eigenvalue of the p-Laplace operator along the Ricci flow on closed manifolds. We show that the first p-eigenvalue is strictly increasing and differentiable almost everywhere along the Ricci flow under some curvature a…
Estimates the first eigenvalue of a Schrödinger operator on minimal submanifolds.
problem Estimating the first eigenvalue of a Schrödinger operator on minimal submanifolds.
method Analyzes the Schrödinger operator L:=−Δ−σ on minimal submanifolds Mn in the unit sphere Sn+m. result Provides an estimate for the first eigenvalue of the Schrödinger operator.
The ball maximizes the first biharmonic Steklov eigenvalue.
problem Maximizing the first biharmonic Steklov eigenvalue for bounded domains.
method Comparing domains with fixed measure to find the maximum eigenvalue.
result The ball maximizes the first positive biharmonic Steklov eigenvalue.
Sharp bounds and rigidity theorems for eigenvalues on manifolds.
problem Estimating eigenvalues and characterizing rigidity on manifolds.
method Volume comparison, Escobar-type eigenvalue comparisons, and Reilly formula.
result Sharp bounds and rigidity conditions for eigenvalues on manifolds.
Sharp bounds found for Steklov-type eigenvalues on surfaces.
problem Finding bounds for the first eigenvalue of Steklov-type problems on compact surfaces.
method Proved bounds using Gaussian curvature constraints and properties of geodesic curvature.
result Sharp lower bounds for the first eigenvalue of Steklov-type problems on compact surfaces.
Study sets lower bounds for Kähler manifolds' Laplacian eigenvalues.
problem Finding bounds for eigenvalues on Kähler manifolds.
method Establishes lower bounds using geometric data like dimension, diameter, and curvature.
result Proves bounds for Laplacian eigenvalues on Kähler manifolds.
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.
Study eigenvalues of p-Laplacian on manifolds with Robin boundary conditions.
problem Investigate the first Robin eigenvalue of the p-Laplacian on Riemannian manifolds.
method Prove eigenvalue comparison theorems and establish sharp bounds.
result Sharp lower bounds for the first Robin eigenvalue of the p-Laplacian.
We prove a Lichnerowicz type lower bound for the first nontrivial eigenvalue of the p-Laplacian on Kähler manifolds. Parallel to the p=2 case, the first eigenvalue lower bound is improved by using a decomposition of the Hessian on Kähler manifolds with positive Ricci curvature.
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.
Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.
problem Finding sharp upper bounds for eigenvalues of magnetic Laplacian.
method Isoperimetric inequalities and bounds in terms of Gaussian curvature.
result Maximal first eigenvalue for geodesic disk on simply connected surfaces.
The study finds bounds for the first eigenvalue of the p-Laplacian on submanifolds.
problem Finding bounds for the first eigenvalue of the p-Laplacian on submanifolds.
method Established an integral inequality for the singular p-laplacian and applied it to submanifolds in the unit sphere.
result Lower bounds for the first eigenvalue of the p-laplacian are obtained for minimal and prescribed scalar curvature submanifolds.
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.
Let (M,g) be an n-dimensional compact Riemannian manifold (n>1) whose metric g(t) evolves by the generalized abstract geometric flow. This paper discusses the evolution, monotonicity and differentiability for the first eigenvalue of the p-Laplacian on (M,g(t)) with respect to time evolution. We prove that t…
Study on eigenvalues of Laplace operator on 1-forms for symmetric spaces.
problem Investigating the first eigenvalue of the Laplace operator on 1-forms in compact inner symmetric spaces.
method Analyzing the Casimir eigenvalue of the highest root for the isotropy representation.
result The first eigenvalue of the Laplace operator on 1-forms is the Casimir eigenvalue of the highest root.
Sharp bounds derived for eigenvalues on specific geometric spaces.
problem Eigenvalue problems on asymptotically hyperbolic manifolds and submanifolds.
method Sharp bounds derived for three types of eigenvalue problems: p-Dirichlet, polyharmonic, and weakly Poincaré-Einstein. result Sharp bounds and their implications for asymptotic sectional curvatures and mean curvature.
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…
We give lower and upper bounds for the first eigenvalue of geodesic balls in spherically symmetric manifolds. These lower and upper bounds are C0-dependent on the metric coefficients. It gives better lower bounds for the first eigenvalue of spherical caps than those from Betz-Camera-Gzyl.
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.
In this paper, we study monotonicity for the first eigenvalue of a class of (p,q)-Laplacian. We find the first variation formula for the first eigenvalue of (p,q)-Laplacian on a closed Riemannian manifold evolving by the Ricci-harmonic flow and construct various monotic quantities by imposing some conditions on ini…
Lower bound found for Steklov eigenvalue on curved manifolds.
problem Finding bounds for Steklov eigenvalues on curved spaces.
method Established a new lower bound using geometric curvature conditions.
result Found a new lower bound for the first non-zero Steklov eigenvalue.
Proves inequality for Steklov eigenvalues in hyperbolic space.
problem Finding bounds for Steklov eigenvalues in hyperbolic geometry.
method Proves isoperimetric inequality for harmonic mean of eigenvalues.
result Establishes inequality for hyperbolic Steklov eigenvalues.
In this note, we obtain the sharp estimates for the first eigenvalue of Paneitz operator for 4-dimensional compact submanifolds in Euclidean space. Since unit spheres and projective spaces can be canonically imbedded into Euclidean space, the corresponding estimates for the first eigenvalue are also obtained.
Estimates eigenvalue for Hermitian manifolds using curvature.
problem Estimating the first eigenvalue of Hermitian manifolds.
method Using holomorphic Ricci and sectional curvatures.
result Established estimates for the first eigenvalue.
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.
Sharp lower bound for p-Laplacian eigenvalue on non-compact manifolds.
problem Estimating eigenvalues of p-Laplacian on non-compact manifolds. method Sharp lower bound established through domain properties and curvature conditions.
result Sharp lower bound for the first Dirichlet eigenvalue of p-Laplacian. In this paper, we successfully generalize the eigenvalue comparison theorem for the Dirichlet p-Laplacian (1<p<∞) obtained by Matei [A.-M. Matei, First eigenvalue for the p-Laplace operator, Nonlinear Anal. TMA 39 (8) (2000) 1051--1068] and Takeuchi [H. Takeuchi, On the first eigenvalue of the p-Laplacian …
Lower bound found for eigenvalue of hypersurface in Riemannian manifold.
problem Finding bounds for eigenvalues of hypersurfaces in Riemannian manifolds.
method Used minimally embedded hypersurface and Ricci curvature constraints.
result Provided a lower bound for the first eigenvalue.