arXiv research

We propose simple conditions equivalent to the discreteness of the spectrum of the Laplace-Beltrami operator on a class of Riemannian manifolds close to warped products. For this class of manifolds we establish a relationship between discreteness of the spectrum and stochastic incompleteness.

Solves geodesics and Laplace-Beltrami spectrum on flag manifolds.

Applying a theorem due to Belopol'ski and Birman, we show that the Laplace-Beltrami operator on 1-forms on ${\bf R}^n$ endowed with an asymptotically Euclidean metric has absolutely continuous spectrum equal to $[0, +\infty)$.

We explicitely compute the essential spectrum of the Laplace-Beltrami operator for $p$-forms for the class of warped product metrics $dσ^2= y^{2a}dy^2 + y^{2b}dθ_{\partial M}^2$, where $y$ is a boundary defining function on a compact manifold with boundary $M$.

Formula derived for Laplace-Beltrami spectrum on homogeneous spaces.

We obtain an asymptotic formula for the eigenvalue distribution function of the Laplace-Beltrami operator on the two-dimensional torus in the adiabatic limit given by a Kronecker foliation. Related problems in number theory are discussed.

We prove a trace formula for three-dimensional spherically symmetric Riemannian manifolds with boundary which satisfy the Herglotz condition: The wave trace is singular precisely at the length spectrum of periodic broken rays. In particular, the Neumann spectrum of the Laplace--Beltrami operator uniquely determines the…

This is an expository article on the question of whether zero lies in the spectrum of the Laplace-Beltrami operator acting on differential forms on a manifold.

On a compact Kähler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eig…

This paper discusses the question whether the discrete spectrum of the Laplace-Beltrami operator is infinite or finite. The borderline-behavior of the curvatures for this problem will be completely determined.

In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds…

We consider non-self-adjoint Schrödinger operators $Δ+V$ where $Δ$ is the Laplace-Beltrami operator on a Zoll manifold $X$ and $V\in C^\infty(X,\mathbb C)$. We obtain asymptotic results on the pseudo-spectrum and numerical range of such operators.

Study spin chains and sigma models on flag manifolds, calculating spectra and geodesics.

Solves sigma model on U(3)/U(1)^3, describing geodesics and spectrum.

The study bounds heat kernel for manifolds with specific curvature conditions.

The paper proves eigenvalues are simple for specific operators on bundles.

Let $(X,h)$ be a compact and irreducible Hermitian complex space of complex dimension $v>1$. In this paper we show that the Friedrichs extension of both the Laplace-Beltrami operator and the Hodge-Kodaira Laplacian acting on functions has discrete spectrum. Moreover we provide some estimates for the growth of the corre…

We prove that the answer to the "zero-in-the-spectrum" conjecture, in its form, suggested by J. Lott, is negative. Namely, we show that for any n > 5 there exists a closed n-dimensional manifold M, so that zero does not belong to the spectrum of the Laplace-Beltrami operator acting on the L^2 forms of all degrees on th…

We prove pointwise bounds for $L^2$ eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with $\mathbb{Q}$-rank one if the corresponding eigenvalues lie below the continuous part of the $L^2$ spectrum. Furthermore, we use these bounds in order to obtain some results concerning the $L^p$ spectrum.

It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m\in\mathbb{N}$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the corresponding Laplace-Beltrami operator $-Δ_M$. In this work we want not only to produce a new type of periodic manifolds …

Paper shows spectra can't distinguish naturally reductive manifolds.

Researchers compute the full spectrum of Laplace operator on distance spheres in symmetric spaces.

Symmetric spaces have unique spectra under certain group actions.

We study spectral properties of the Laplace-Beltrami operator on two relevant almost-Riemannian manifolds, namely the Grushin structures on the cylinder and on the sphere. This operator contains first order diverging terms caused by the divergence of the volume. We get explicit descriptions of the spectrum and the eige…

The paper proves a bound on eigenvalues for surfaces embedded in 3D space.

This paper is devoted to the study of the conformal spectrum (and more precisely the first eigenvalue) of the Laplace-Beltrami operator on a smooth connected compact Riemannian surface without boundary, endowed with a conformal class. We give a constructive proof of a critical metric which is smooth except at some coni…

In this paper we obtain lower bound estimates of the spectrum of Laplace-Beltrami operator on complete submanifolds with bounded mean curvature, whose ambient space admits a Riemannian submersion over a Riemannian manifold with negative sectional curvature. Our main theorem generalizes many previous known estimates and…

To what extent does the eigenvalue spectrum of the Laplace-Beltrami operator on a compact Riemannian manifold determine the geometry of the manifold? We give examples of isospectral manifolds with different local geometry including continuous families of isospectral negatively curved manifolds with boundary as well as …

We consider a family of compact manifolds which shrinks with respect to an appropriate parameter to a graph. The main result is that the spectrum of the Laplace-Beltrami operator converges to the spectrum of the (differential) Laplacian on the graph with Kirchhoff boundary conditions at the vertices. On the other hand,…

We prove a Hardy inequality for uniformly elliptic operators subject to Dirichlet or mixed boundary conditions on domains $Ω$ with piecewiese smooth boundary in arbitrary Riemannian Manifolds (M, g). Employing an approach of E.B. Davies for the euclidean case, we show that it implies a sufficient geometric criterion un…

The paper bounds eigenvalues of hyperbolic manifolds with infinite volume.

We study the $L^p$-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces $M=Γ\backslash X$ with finite volume and arithmetic fundamental group $Γ$ whose universal covering $X$ is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces c…

We study the spectral geometric properties of the scalar Laplace-Beltrami operator associated to the Weil-Petersson metric $g_{\mathrm{WP}}$ on $\mathcal M_γ$, the Riemann moduli space of surfaces of genus $γ> 1$. This space has a singular compactification with respect to $g_{\mathrm{WP}}$, and this metric has crossing…

Eigenvalues of manifolds with cylindrical boundaries approximated by graph Laplacians.

We establish a lower bound for the real eigenvalues of a Laplace-Beltrami operator with an $L^\infty$-drift term. We make no assumptions that the operator is self-adjoint or that the drift has any additional regularity. In the case where the operator is self-adjoint, this establishes a lower bound on the spectrum witho…

Graphs approximate Laplacian spectra on manifolds.

We establish an explicit expression for the smallest non-zero eigenvalue of the Laplace--Beltrami operator on every homogeneous metric on the 3-sphere, or equivalently, on SU(2) endowed with left-invariant metric. For the subfamily of 3-dimensional Berger spheres, we obtain a full description of their spectra. We also …

A new method matches similar regions in non-rigid shapes using spectra of differential operators.

Formula for Laplace-Beltrami on orthogonal group in Euclidean coords.

Formula derived for Laplace-Beltrami on Stiefel manifold.

The paper studies Kähler metrics from finite Monge-Ampère mass exhaustion functions.

We prove a lower bound for the $k$-th Steklov eigenvalues in terms of an isoperimetric constant called the $k$-th Cheeger-Steklov constant in three different situations: finite spaces, measurable spaces, and Riemannian manifolds. These lower bounds can be considered as higher order Cheeger type inequalities for the Ste…

We consider the Dirichlet Laplacian in unbounded strips on ruled surfaces in any space dimension. We locate the essential spectrum under the condition that the strip is asymptotically flat. If the Gauss curvature of the strip equals zero, we establish the existence of discrete spectrum under the condition that the curv…

Method finds domain of Laplace-Beltrami operator on 2D almost-Riemannian manifolds.

Study of resonances and residue operators for hyperbolic spaces.

Two-dimensional almost-Riemannian structures are generalized Riemannian structures on surfaces for which a local orthonormal frame is given by a Lie bracket generating pair of vector fields that can become collinear. Generically, the singular set is an embedded one dimensional manifold and there are three type of point…

We prove trace identities for commutators of operators, which are used to derive sum rules and sharp universal bounds for the eigenvalues of periodic Schroedinger operators and Schroedinger operators on immersed manifolds. In particular, we prove bounds on the eigenvalue lambda_{N+1} in terms of the lower spectrum, bou…

In this paper, we construct Laplace-Beltrami operators associated with arbitrary Riemannian metrics on noncommutative tori of any dimension. These operators enjoy the main properties of the Laplace-Beltrami operators on ordinary Riemannian manifolds. The construction takes into account the non-triviality of the group o…