arXiv research

Sharp Steklov eigenvalue estimates for differential forms on manifolds.

The paper connects eigenvalue problems for various operators and establishes inequalities and asymptotic formulas for heat traces.

Paper derives a Reilly type integral formula and applies it to inequalities and eigenvalue problems.

Study on eigenvalues of complex Hessian operator on pseudoconvex manifolds.

In this paper, we extend the Reilly formula for drifting Laplacian operator and apply it to study eigenvalue estimate for drifting Laplacian operators on compact Riemannian manifolds boundary. Our results on eigenvalue estimates extend previous results of Reilly and Choi and Wang.

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…

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…

Variational methods yield formulas for eigenvalues of elliptic operators, with applications to metric evolution.

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…

Study sharp lower bounds on negative eigenvalues of magnetic Pauli operator.

Paper derives second variation formula for eigenvalue functionals on surfaces.

Derives integral formula for differential forms on compact spaces with applications.

Derives formulas for differential forms on weighted manifolds.

In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an $n$-dimensional Riemannian manifold.

Sharp bounds and rigidity theorems for eigenvalues on manifolds.

We discuss asymptotic behavior of the eigenvalue distribution of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions normal to the leaves (in the adiabatic limit). Motivated by analogies with semiclassical spectral asymptotics, we use ideas…

The paper studies eigenvalues in gaps of the essential spectrum of a Bochner-Schrödinger operator.

Let $G/K$ be a simply connected spin compact inner irreducible symmetric space, endowed with the metric induced by the Killing form of $G$ sign-changed. We give a formula for the square of the first eigenvalue of the Dirac operator in terms of a root system of $G$. As an example of application, we give the list of the …

Three counterexamples show higher eigenvalue multiplicities than conjectured.

Eigenvalues of random hyperbolic surface covers converge to hyperbolic plane's.

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 $n = 7$.

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

Researchers improve spectrum reconstruction formula with proof.

For an $n$-dimensional polytope $Ω$ in $\mathbb{R}^{n}$, we study lower bounds for eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. In the asymptotic formula on the average of the first $k$ eigenvalues, Li and Yau (1983) obtained the first term with the order $k^{\frac2n}$, which is optimal. The next l…

In this paper, we obtain "universal" inequalities for eigenvalues of the weighted Hodge Laplacian on a compact self-shrinker of Euclidean space. These inequalities generalize the Yang-type and Levitin-Parnovski inequalities for eigenvalues of the Laplacian and Laplacian. From the recursion formula of Cheng and Yang \ci…

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, 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 give a formula for the first eigenvalue of the Dirac operator acting on spinor fields of a spin compact irreducible symmetric space $G/K$.

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigen…

For a bounded domain $Ω$ with a piecewise smooth boundary in a complete Riemannian manifold $M$, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of $L^2(Ω)$ in place of the Rayleigh-Ritz formula, we obtain inequalities for …

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…

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 …

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 obtain a new abstract formula relating eigenvalues of a self-adjoint operator to two families of symmetric and skew-symmetric operators and their commutators. This formula generalizes earlier ones obtained by Harrell, Stubbe, Hook, Ashbaugh, Hermi, Levitin and Parnovski. We also show how one can use t…

The paper finds a lower bound for a Neumann eigenvalue of surfaces with flat ends.

Eigenvalue estimates for Hodge Laplacian on Fano manifolds lead to geometric insights.

This paper studies eigenvalues of the clamped plate problem on a bounded domain in an $n$-dimensional Euclidean space. We give an estimate for the gap between $\sqrt {Γ_{k+1}-Γ_{1}}$ and $\sqrt {Γ_{k}-Γ_{1}}$, for any positive integer $k$. According to the asymptotic formula of Agmon and Pleijel, we know, the gap betwe…

Harer-Zagier formulas generalized to knot matrix models.

The paper studies eigenvalue inequalities for a clamped plate problem involving a generalized elliptic differential operator.

Study eigenvalues of CROSS spaces with various metrics.

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…

Paper studies eigenvalues of a specific operator on Riemannian manifolds.

Given a compatible vector field on a compact connected almost-complex manifold, we show in this article that the multiplicities of eigenvalues among the zero point set of this vector field have intimate relations. We highlight a special case of our result and reinterpret it as a vanishing-type result in the framework o…

Contact Riemannian manifolds, with not necessarily integrable complex structures, are the generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection on such a manifold plays the role of Tanaka-Webster connection in the pseudohermitian case. We prove the contact Riemannian version of…

Study on symplectic Dirac operators on foliations, estimating eigenvalues.

In this work we generalise various recent results on the evolution and monotonicity of the eigenvalues of certain geometric operators under specified geometric flows. Given a closed, compact Riemannian manifold $\big(M^n,g(t)\big)$ and a smooth function $η\in C^{\infty}(M)$ we consider the family of operators $\mathbb{…

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…

Formulas for spectra of higher spin operators on sphere subbundles.