arXiv research

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

Sphere theorems for p-Laplacian eigenvalues 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 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…

Let $Ω$ be a star-shaped bounded domain in $(\mathbb{S}^{n}, ds^{2})$ with smooth boundary. In this article, we give a sharp lower bound for the first non-zero eigenvalue of the Steklov eigenvalue problem in $Ω.$ This result is the generalization of a result given by Kuttler and Sigillito for a star-shaped bounded doma…

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

The paper solves conditions for prescribing scalar and Gauss curvatures on manifolds with zero first eigenvalue.

The study analyzes weighted manifolds with curvature bounds, proving eigenvalue estimates and inequalities.

Improved lower bound for the first eigenvalue of embedded minimal hypersurfaces in the unit sphere

Lower bound found for Steklov eigenvalue on curved manifolds.

Using expander graphs, we construct a sequence of smooth compact surfaces with boundary of perimeter N, and with the first non-zero Steklov eigenvalue uniformly bounded away from zero. This answers a question which was raised in [9]. The genus grows linearly with N, this is the optimal growth rate.

We define the distance between edges of graphs and study the coarse Ricci curvature on edges. We consider the Laplacian on edges based on the Jost-Horak's definition of the Laplacian on simplicial complexes. As one of our main results, we obtain an estimate of the first non-zero eigenvalue of the Laplacian by the Ricci…

We discuss optimal lower bounds for eigenvalues of Laplacians on weighted graphs. These bounds are formulated in terms of the geometry and, more specifically, the inradius of subsets of the graph. In particular, we study the first non-zero eigenvalue in the finite volume case and the first eigenvalue of the Dirichlet L…

We give an estimate on the lower bound of the first non-zero eigenvalue of the Laplacian for a closed Riemannian manifold with positive Ricci curvature in terms of the in-diameter and the lower bound of the Ricci curvature.

Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.

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…

Upper bounds for magnetic Laplacian eigenvalues on planar domains.

Upper bounds for Steklov eigenvalues on curved submanifolds.

We establish lower bounds for the first non-zero eigenvalue for the natural geometric sub-elliptic Laplacian operator defined on sub-Riemannian manifolds of step 2 that satisfy a positive curvature condition. The methods are very general and can be applied even when the sub-Riemannian geometry has considerable torsion.

We extend the results given by Colbois, Dryden and El Soufi on the relationships between the eigenvalues of the Laplacian and an extrinsic invariant called intersection index, in two directions. First, we replace this intersection index by invariants of the same nature which are stable under small perturbations. Second…

Along the line of the Yang Conjecture, we give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with negative lower bound of Ricci curvature in terms of the in-diameter and the lower bound of Ricci curvature.

Paper confirms Yau's conjecture about sphere eigenvalues.

We consider a solution f of a certain Dirichlet Problem on a domain in $S^{(n+1)}$ whose boundary is a minimal hypersurface and we prove a Poincare type inequality for f. One have equality iff Yau's conjecture about the first non-zero eigenvalue of closed minimal hypersurfaces of $S^{(n+1)}$ is true.

Study on shape optimization for specific eigenvalue problems on domains.

New metrics on 3D manifolds with large Steklov eigenvalues.

Study shows optimal spectral gaps diminish in large genus surfaces.

In this paper we will prove new extrinsic upper bounds for the eigenvalues of the Dirac operator on an isometrically immersed surface $M^2 \hookrightarrow {\Bbb R}^3$ as well as intrinsic bounds for 2-dimensional compact manifolds of genus zero and genus one. Moreover, we compare the different estimates of the eigenval…

We study the small eigenvalues of the Hodge Laplacian on collaping torus bundles with bounded curvature. In the first part of this dissertation, we consider examples of bundles on S^1 and T^2 with homogeneous structure. In the second part, we give a lower bound of the first non-zero eigenvalue of the 1-form Laplacian o…

In this paper, we study the evolving behaviors of the first eigenvalue of Laplace-Beltrami operator under the normalized backward Ricci flow, construct various quantities which are monotonic under the backward Ricci flow and get upper and lower bounds. We prove that in cases where the backward Ricci flow converges to a…

Improved estimate for eigenvalues of minimal hypersurfaces in spheres.

Paper provides lower bounds for eigenvalues on singular Riemannian foliations.

} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $Δ^2$ on a bounded smooth domain $\Om$ in the Euclidean $n$-space ${\bf R}^n$ ($n\ge2$) and then prove that the corresponding first non-zero eigenvalue $Υ_1(\Om)$ admits the isoperimetric inequality of Szegö-Weinberger type: $Υ_…

We generalise a theorem of Engman and Abreu--Freitas on the first invariant eigenvalue of non-negatively curved $S^{1}$-invariant metrics on $\mathbb{CP}^{1}$ to general toric Kähler metrics with non-negative scalar curvature. In particular, a simple upper bound of the first non-zero invariant eigenvalue for such metri…

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non zero eigenvalue of the drift Lapl…

We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take th…

Study on stability of quaternion-Kähler manifolds using eigenvalue estimates.

Let M^n be a compact n-dimensional principal T^k-bundle. We consider collapsings of M on N=M/T^k such that the diameter and sectional curvature of M satisfy diam(M)<d and |K(M)|<a, and give examples of collapsings for all k such that the first non-zero eigenvalue of Laplacian acting on 1-forms and 2-forms of M are boun…

We show that zero is not an eigenvalue of the conformal Laplacian for generic Riemannian metrics. We also discuss non-compactness for sequences of metrics with growing number of negative eigenvalues of the conformal Laplacian.

We prove a lower bound estimate for the first non-zero eigenvalue of the Witten-Laplacian on compact Riemannian manifolds. As an application, we derive a lower bound estimate for the diameter of compact gradient shrinking Ricci solitons. Our results improve some previous estimates which were obtained by the first autho…

In recent years, eigenvalue optimization problems have received a lot of attention, in particular, due to their connection with the theory of minimal surfaces. In the present paper we prove that on any orientable surface there exists a smooth metric maximizing the first normalized Steklov eigenvalue. For surfaces of ge…

Example surfaces with curvature bounds but no Laplacian eigenvalue lower bound.

Extends Choi-Wang inequality to Li-Xia affine connections.

We prove a new upper bound for the first eigenvalue of the Dirac operator of a compact hypersurface in any Riemannian spin manifold carrying a non-trivial twistor spinor without zeros on the hypersurface. The upper bound is expressed as the first eigenvalue of a drifting Schrödinger operator on the hypersurface. Moreov…

We consider the Jacobi operator, defined on a closed oriented hypersurfaces immersed in the Euclidean space with the same volume of the unit sphere. We show a local generalization for the classical result of the Willmore functional for the Euclidean sphere. As a consequence, we prove that the first eigenvalue of the Ja…

In this paper, we numerically investigate the length spectra and the low-lying eigenvalue spectra of the Laplace-Beltrami operator for a large number of small compact(closed) hyperbolic (CH) 3-manifolds. The first non-zero eigenvalues have been successfully computed using the periodic orbit sum method, which are compar…

The study finds lower bounds for the first eigenvalue of the Laplacian in planar domains with magnetic fields.

In this paper we study the smallest non-zero eigenvalue $λ_1$ of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for $λ_1$ in terms of moment polytope data. We show that this bound can only be attained for $\mathbb{CP}^n$ endowed with the Fubini-Study metric and therefore $\mathbb{CP}^n$ endowe…

Study eigenvalues of Laplace operator on specific 3D manifolds under Ricci flow.