Eigenvalue pinching on $\text{spin}^c$ manifolds

Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound

We revisit classical eigenvalue inequalities due to Buser, Cheng, and Gromov on closed Riemannian manifolds, and prove the versions of these results for the Dirichlet and Neumann boundary value problems. Eigenvalue multiplicity bounds and related open problems are also discussed.

The First Dirac Eigenvalue on Manifolds with Positive Scalar Curvature

We show that on every compact spin manifold admitting a Riemannian metric of positive scalar curvature Friedrich's eigenvalue estimate for the Dirac operator can be made sharp up to an arbitrarily small given error by choosing the metric suitably.

Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds

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 upper bound for Steklov eigenvalues. A Pohozaev-type identity for harmonic functions on the domain and the min-max variational characterization of both eigenvalues are important ingredients.

带圆柱边界流形上的狄利克雷特征值近似

Dirichlet Eigenvalue Approximation on Manifolds with Cylindrical Boundary

研究通过构造(ε,ρ)近邻图的截断图拉普拉斯算子谱来近似流形上的狄利克雷特征值，证明在满足曲率、注入半径和直径约束的流形类中，特征值收敛性。

Lower bounds for the first eigenvalue of $p$-Laplacian on Kähler manifolds

We study the eigenvalue problem for the $p$-Laplacian on Kähler manifolds. Our first result is a lower bound for the first nonzero eigenvalue of the $p$-Laplacian on compact Kähler manifolds in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature for $p\in (1, 2]$. Our second result is a sharp lower bound for the first Dirichlet eigenvalue of the $p$-Laplacian on compact Kähler manifolds with smooth boundary for $p\in (1, \infty)$. Our results generalize corresponding results for the Laplace eigenvalues on Kähler manifolds proved in […

Eigenvalues on Spherically Symmetric Manifolds

In this article we will explore Dirichlet Laplace eigenvalues on balls on spherically symmetric manifolds. We will compare any Dirichlet Laplace eigenvalue with the corresponding Dirichlet Laplace eigenvalue on balls in Euclidean space with the same radius. As a special case we will show that the Dirichlet Laplace eigenvalues on balls with small radius on the sphere are smaller than the corresponding eigenvalues on the Euclidean ball with the same radius. While the opposite is true for the Dirichlet Laplace eigenvalues of hyperbolic spaces.

Lower bounds for the first eigenvalue of the Laplacian on Kähler manifolds

We establish lower bound for the first nonzero eigenvalue of the Laplacian on a closed Kähler manifold in terms of dimension, diameter, and lower bounds of holomorphic sectional curvature and orthogonal Ricci curvature. On compact Kähler manifolds with boundary, we prove lower bounds for the first nonzero Neumann or Dirichlet eigenvalue in terms of geometric data. Our results are Kähler analogues of well-known results for Riemannian manifolds.

Upper and lower bounds for the first eigenvalue and the volume entropy of noncompact Kähler manifolds

We find upper and lower bounds for the first eigenvalue and the volume entropy of a noncompact real analytic Kähler manifold, in terms of Calabi's diastasis function and diastatic entropy, which are sharp in the case of the complex hyperbolic space. As a corollary we obtain explicit lower bounds for the first eigenvalue of the geodesic balls of an Hermitian symmetric space of noncompact type.

Sharp modulus of continuity for parabolic equations on manifolds and lower bounds for the first eigenvalue

We derive sharp estimates on modulus of continuity for solutions of the heat equation on a compact Riemannian manifold with a Ricci curvature bound, in terms of initial oscillation and elapsed time. As an application, we give an easy proof of the optimal lower bound on the first eigenvalue of the Laplacian on such a manifold as a function of diameter.

Eigenvalues for a Schrodinger operator on a closed Riemannian manifold with holes

In this article we consider a closed Riemannian manifold (M,g) and A a subset of M. The purpose of this article is the comparison between the eigenvalues of a Schrodinger operator on the manifold (M,g) and the eigenvalues on the manifold (M-A,g) with Dirichlet boundary conditions.

Noncompact complete Riemannian manifolds with dense eigenvalues embedded in the essential spectrum of the Laplacian

We prove sharp criteria on the behavior of radial curvature for the existence of asymptotically flat or hyperbolic Riemannian manifolds with prescribed sets of eigenvalues embedded in the spectrum of the Laplacian. In particular, we construct such manifolds with dense embedded point spectrum and sharp curvature bounds.

Comparison Results, Exit Time Moments, And Eigenvalues On Riemannian Manifolds With A Lower Ricci Curvature Bound

We study the relationship between the geometry of smoothly bounded domains in complete Riemannian manifolds and the associated sequence of $L^1$-norms of exit time moments for Brownian motion. We establish bounds for Dirichlet eigenvalues and, for closed manifolds, we establish a comparison result for elements of the moment sequence.

Eigenvalue distribution for non-self-adjoint operators on compact manifolds with small multiplicative random perturbations

In this work we extend a previous work about the Weyl asymptotics of the distribution of eigenvalues of non-self-adjoint differential operators with small multiplicative random perturbations, by treating the case of operators on compact manifolds

Finiteness of Small Eigenvalues of Geometrically Finite Rank one Locally Symmetric Manifolds

Let M be a geometrically finite rank one locally symmetric manifolds. We prove that the spectrum of the Laplace operator on M is finite in a small interval which is optimal.

Eigenvalues in Spectral Gaps of a Perturbed Periodic Manifold

We consider a non-compact Riemannian periodic manifold such that the corresponding Laplacian has a spectral gap. By continuously perturbing the periodic metric locally we can prove the existence of eigenvalues in a gap. A lower bound on the number of eigenvalue branches crossing a fixed level is established in terms of a discrete eigenvalue problem. Furthermore, we discuss examples of perturbations leading to infinitely many eigenvalue branches coming from above resp. finitely many branches coming from below.