arXiv research

Variational method for eigenvalues on manifolds.

Sharp bounds derived for the first two Steklov eigenvalues of exterior domains.

New framework for studying eigenvalue functionals of metrics.

Paper derives second variation formula for eigenvalue functionals on surfaces.

Upper bound found for Steklov eigenvalues counting function.

The paper finds lower bounds for the first eigenvalue of p-Laplacian in specific manifolds.

Paper finds inequalities for eigenvalues of buckling problems on special metric spaces.

In [LS], it is shown shown that the first eigenvalue of the Laplacian restricted to the space of invariant functions on a toric Kähler manifold (i.e. $λ_1^\mathbb{T}$, the invariant first eigenvalue) is an unbounded function of the toric Kähler metric. In this note we show that, seen as a function on the space of toric…

In this paper, we investigate critical points of the Laplacian's eigenvalues considered as functionals on the space of Riemmannian metrics or a conformal class of metrics on a compact manifold. We obtain necessary and sufficient conditions for a metric to be a critical point of such a functional. We derive specific con…

Eigenvalue bounds for Schrödinger operators on Ricci shrinkers and related manifolds.

In this paper, we study monotonicity formulas of eigenvalues and entropies along the rescaled List's extended Ricci flow. We derive some monotonicity formulas of eigenvalues of Laplacian which generalize those of Li in [8] and Cao-Hou-Ling in [3]. Moreover, we also consider monotonicity formulas of $\mathcal{F}_k$-func…

Study on eigenvalues of Laplace operator on 1-forms for symmetric spaces.

Eigenvalue estimates for Beltrami-Laplacian under specific curvature conditions.

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…

Upper bounds for Steklov eigenvalues of warped products are derived.

This paper improves lower bounds for the first nonzero Steklov eigenvalue on compact manifolds.

Study calculates Hessian of Busemann function on Damek-Ricci spaces.

We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degener…

We investigate the second Dirac eigenvalue on Riemannian manifolds admitting a Killing spinor. In small dimensions the whole Dirac spectrum depends on special eigenvalues on functions and 1-forms. We compute and discuss the formulas in dimension $n = 7$.

Study maximizes eigenvalues in dimensions 3 and above.

The Riemann curvature tensor is a central mathematical tool in Einstein's theory of general relativity. Its related eigenproblem plays an important role in mathematics and physics. We extend M-eigenvalues for the elasticity tensor to the Riemann curvature tensor. The definition of M-eigenproblem of the Riemann curvatur…

New method to bound Laplacian eigenvalues of geodesic balls.

Study approximates product of spheres using Laplacian eigenvalues.

In this study, we attempted to determine how eigenvalues change, according to random matrix theory (RMT), in stock market data as the number of stocks comprising the correlation matrix changes. Specifically, we tested for changes in the eigenvalue properties as a function of the number and type of stocks in the correla…

Paper proves surfaces with specific symmetries have the first Steklov eigenvalue.

We study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Ricci tensor. We prove an existence theorem for a wide class of symmetric functions on manifolds with positive Ricci curvature, provided the conformal class admits an admissible metric.

Derives a formula for the second variation of the Laplace eigenvalue functional on manifolds.

Study eigenvalues of conformal Laplacian under Sire-Xu normalization.

The paper finds extremum values for mixed Laplacian eigenvalues on triangles and trapezoids.

Kernel methods are successful approaches for different machine learning problems. This success is mainly rooted in using feature maps and kernel matrices. Some methods rely on the eigenvalues/eigenvectors of the kernel matrix, while for other methods the spectral information can be used to estimate the excess risk. An …

On 7D quaternionic contact manifolds, eigenvalue bounds imply special structure.

We investigate the statistical properties of the correlation matrix between individual stocks traded in the Korean stock market using the random matrix theory (RMT) and observe how these affect the portfolio weights in the Markowitz portfolio theory. We find that the distribution of the correlation matrix is positively…

Study eigenvalues on quaternion-Kähler manifolds with geometric bounds.

Given a compact Riemannian manifold (M n , g) with boundary $\partial$M , we give an estimate for the quotient $\partial$M f d$μ$ g M f d$μ$ g , where f is a smooth positive function defined on M that satisfies some inequality involving the scalar Laplacian. By the mean value lemma established in [37], we provide a dif…

Sharp Steklov eigenvalue estimates for differential forms on manifolds.

The paper studies eigenvalues of Xin-Laplacian on Riemannian manifolds.

Eigenvalue problem for Kähler metrics on compact manifolds.

In this paper, we extend the Witten-Helffer-Sjöstrand theory from Morse functions to generalized Morse functions. In this case, the spectrum of the Witten deformed Laplacian $Δ(t)$, for large t, can be seperated into the small eigenvalues (which tend to 0 as $t\rightarrow\infty$), large and very large eigenvalues (both…

Paper derives inequalities for eigenvalues of Witten-Laplacian under fixed volume constraint.

The i-th eigenvalue of the Laplacian on a surface can be viewed as a functional on the space of Riemannian metrics of fixed area. Extremal points of these functionals correspond to surfaces admitting minimal isometric immersions into spheres. Recently, critical metrics for the first eigenvalue were classified on tori a…

Study on new Monge-Ampère functionals and their variational problems.

In this paper we study the problem of conformally deforming a metric to a prescribed symmetric function of the eigenvalues of the Schouten tensor on compact Riemannian manifolds with boundary. We prove its solvability and the compactness of the solution set, provided the Ricci tensor is non-negative definite.

The paper reformulates Bakry-Émery curvature on graphs using eigenvalues.

Paper calculates eigenvalue decay rates for neural network kernels on general domains.

We study a Dirichlet-to-Neumann eigenvalue problem for differential forms on a compact Riemannian manifold with smooth boundary. This problem is a natural generalization of the classical Steklov problem on functions. We derive a number of upper and lower bounds for the first eigenvalue in several contexts: many of thes…

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 $C^r$ Riemannian …

Proves existence of maximizers for eigenvalue optimization on manifolds.

Eigenvalues on spheres are compared to the unit round sphere, proving a sharp bound and equality condition.