arXiv research

Introduces a new elliptic operator with positive eigenvalue.

Estimates eigenvalues of poly-Laplace operator on lattice subgraphs.

The paper studies graph Laplace operator behavior near isolated singularities.

The standard Laplace operator is a generalization of the Hodge Laplace operator on differential forms to arbitrary geometric vector bundles, alternatively it can be seen as generalization of the Casimir operator acting on sections of homogeneous vector bundles over symmetric spaces to general Riemannian manifolds. Stre…

Graph Laplace operators uniquely identify metrics and densities on manifolds.

Formula derived for Laplace-Beltrami on Stiefel manifold.

The paper introduces Laplace-type operators for functions defined on the tangent space of a Finsler Lie algebroid, using a volume form on the prolongation of the algebroid. It also presents the construction of a horizontal Laplace operator for forms defined on the prolongation of the algebroid. All of the Laplace opera…

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

We show that eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a Riemannian manifold are approximated by eigenvalues and eigenvectors of a (suitably weighted) graph Laplace operator of a proximity graph on an epsilon-net.

This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.

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

The higher spin Laplace operator has been constructed recently as the generalization of the Laplacian in higher spin theory. This acts on functions taking values in arbitrary irreducible representations of the Spin group. In this paper, we first provide a decomposition of the higher spin Laplace operator in terms of Rr…

Paper studies Laplace operator estimates in harmonic map heat flows.

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…

Algorithm computes eigenvalues and eigenforms on Calabi-Yau threefolds.

We study the asymptotic behaviour of regularized determinants of certain Laplace type operators with respect to singular deformations of the underlying manifold which are obtained by stretching a tubular neighborhood of an embedded separating hypersurface to a cylinder of infinite length. Using the asymptotic expansion…

Extends geometric structures to manifolds with new operators.

We study the higher spin Dirac operators on 3-dimensional manifolds and show that there exist two Laplace type operators for each associated bundle. Furthermore, we give lower bound estimations for the first eigenvalues of these Laplace type operators.

We give a new definition of a Laplace operator for Finsler metric as an average with regard to an angle measure of the second directional derivatives. This definition uses a dynamical approach due to Foulon that does not require the use of connections nor local coordinates. We show using 1-parameter families of Katok--…

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

For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differ…

Solves geodesics and Laplace-Beltrami spectrum on flag manifolds.

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

Trapezoids uniquely identified by their Dirichlet Laplace spectrum.

We consider odd Laplace operators acting on densities of various weight on an odd Poisson (= Schouten) manifold $M$. We prove that the case of densities of weight 1/2 (half-densities) is distinguished by the existence of a unique odd Laplace operator depending only on a point of an ``orbit space'' of volume forms. This…

We establish the existence of analytic curves of eigenvalues for the Laplace-Neumann operator through an analytic variation of the metric of a compact Riemannian manifold $M$ with boundary by means of a new approach rather than Kato's method for unbounded operators. We obtain an expression for the derivative of the cur…

Study submanifolds in hyperbolic space, focusing on their boundary and Laplace operator.

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…

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

This thesis covers different aspects of the p-Laplace operators on Riemannian manifolds. Chapter 2. Potential theoretic aspects: the Khasmkinskii condition. Chapter 3: sharp eigenvalue estimates with Ricci curvature lower bounds. Chapter 4: Critical sets of (2-)harmonic functions.

We consider generalized Hodge-Laplace operators $αd δ+ βδd$ for $α, β> 0$ on $p$-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.

We define a discrete Laplace-Beltrami operator for simplicial surfaces. It depends only on the intrinsic geometry of the surface and its edge weights are positive. Our Laplace operator is similar to the well known finite-elements Laplacian (the so called ``cotan formula'') except that it is based on the intrinsic Delau…

Given a Hodge manifold, it is introduced a self-adjoint operator on the space of endomorphisms of the global holomorphic sections of the polarization line bundle. Such operator is shown to approximate the Laplace operator on functions when composed with Berezin-Toeplitz quantization map and its adjoint up to an error w…

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 $[0, +\infty)$.

Let $P$ be a non-negative self-adjoint Laplace type operator acting on sections of a hermitian vector bundle over a closed Riemannian manifold. In this paper we review the close relations between various $P$-related coefficients such as the mollified spectral counting coefficients, the heat trace coefficients, the reso…

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.

In the context of synthetic differential geometry, we study the Laplace operator an a Riemannian manifold. The main new aspect is a neighbourhood of the diagonal, smaller than the second neighbourhood usually required as support for second order differential operators. The new neighbourhood has the property that a func…

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.

Let $x : M \to E^m$ be an isometric immersion of a Riemannian manifold $M$ into a Euclidean $m$-space. Denote by $Δ$ the Laplace operator of $M$. Then $Δ$ gives rise to a differentiable map $L :M \to E^m$, called the Laplace map, defined by $L(p)=(Δx)(p)$, $p\in M$. We call $L(M)$ the Laplace image, and the transformat…

For a finite reflection subgroup $G\leq O(n+1,1,\mR)$ of the conformal group of the sphere with standard conformal structure $(S^n,[g_0])$, we geometrically derive differential-difference Dunkl version of the series of conformally invariant differential operators with symbols given by powers of Laplace operator. The co…

The moduli space NK of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6…

The known upper bounds for the multiplicities of the Laplace-Beltrami operator eigenvalues on the real projective plane are improved for the eigenvalues with even indexes. Upper bounds for Dirichlet, Neumann and Steklov eigenvalues on the real projective plane with holes are also provided.

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

Bounds on spectral gaps of hyperbolic 3-manifolds and orbifolds.

It is shown that the estimates obtained by Manfredo P. do Carmo and Detang Zhou, in their paper "Eigenvalue estimate on complete noncompact Riemannian manifolds and applications", for the first eigenvalue of the Laplace-Beltrami operator on open manifolds, via an oscillation theorem, can be naturally extended for the s…

Symmetry proven for positive solutions of a weighted p-Laplace operator inequality.

In this paper are given explicit calculations of Laplace operator spectrum for smooth real/complex-valued functions on all connected compact simple rank three Lie groups with biinvariant Riemannian metric and established a connection of obtained formulas with the number theory and integer ternary and binary quadratic f…

The purpose of this note is to provide a short cut presentation of a Mayer-Vietoris formula due to Burghelea-Friedlander-Kappeler for the regularized determinant in the case of elliptic operators of Laplace Beltrami type in the form typically needed in applications to torsion.