Laplace-Beltrami Operator Research Articles

Rotational hypersurfaces in Lorentz-Minkowski 4-space

In this study, we study rotational hypersurfaces in 4-dimensional Lorentz-Minkowski space. We find the rotational hypersurfaces about spacelike axis according to Gaussian and mean curvatures in $E_{1}^{4}$ and give some results with the aid of the Gaussian and mean curvatures. After that, we deal with the Gauss map of rotational hypersurface about spacelike axis by obtaining the Gaussian and mean curvatures. We obtain the second and third Laplace-Beltrami operators on rotational hypersurface about spacelike axis in $E_{1}^{4}$. Also, we give these characterizations for rotational hypersurfaces about timelike and lightlike axes, too.

On Hardy-Littlewood-Pólya and Taikov type inequalities for multiple operators in Hilbert spaces

We present unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Poly\'a and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemmanian manifolds and derive the well-known Taikov and Hardy-Littlewood-Poly\'a inequalities for functions defined on $d$-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of one class by elements of other class. In addition, we establish sharp Solyar-type inequalities for unbounded closed operators with closed range.

Leibniz cohomology and connections on differentiable manifolds

We show how an affine connection on a Riemannian manifold occurs naturally as a cochain in the complex for Leibniz cohomology of vector fields with coefficients in the adjoint representation. The Leibniz coboundary of the Levi-Civita connection can be expressed as a sum of two terms, one the Laplace-Beltrami operator and the other a Ricci curvature term. The vanishing of this coboundary has an interpretation in terms of eigenfunctions of the Laplacian. Separately, we compute the Leibniz cohomology with adjoint coefficients for a certain family of vector fields on Euclidean Rn corresponding to the affine orthogonal Lie algebra, n≥3.

Continuous Fuzzy Transform as Integral Operator

The fuzzy transform (F-transform) is ubiquitous in different research fields and applications, such as image and data compression, data mining, knowledge discovery, and the analysis of linguistic expressions. As a generalization of the F-transform, in this article, we introduce the continuous F-transform and its inverse, as an integral operator induced by a kernel function. Through the relation between membership functions and integral kernels, we show that the main properties (e.g., continuity and symmetry) of the membership functions are inherited by the continuous F-transform. Then, the relation between the continuous F-transform and integral operators is used to introduce a data-driven F-transform, which encodes intrinsic information (e.g., structure, geometry, and sampling density) about the input data. In this way, we avoid coarse fuzzy partitions, which group data into large clusters that do not adapt to their local behavior, or a too dense fuzzy partition, which generally has cells that are not covered by the data, thus being redundant and resulting in a higher computational cost. To this end, the data-driven membership functions are defined by properly filtering the spectrum of the Laplace–Beltrami operator associated with the input data. Finally, we introduce the space of continuous F-transforms, which is useful for the comparison of different continuous F-transforms and for their efficient computation.

Manifold Learning with Arbitrary Norms

Manifold learning methods play a prominent role in nonlinear dimensionality reduction and other tasks involving high-dimensional data sets with low intrinsic dimensionality. Many of these methods are graph-based: they associate a vertex with each data point and a weighted edge with each pair. Existing theory shows that the Laplacian matrix of the graph converges to the Laplace-Beltrami operator of the data manifold, under the assumption that the pairwise affinities are based on the Euclidean norm. In this paper, we determine the limiting differential operator for graph Laplacians constructed using $\textit{any}$ norm. Our proof involves an interplay between the second fundamental form of the manifold and the convex geometry of the given norm's unit ball. To demonstrate the potential benefits of non-Euclidean norms in manifold learning, we consider the task of mapping the motion of large molecules with continuous variability. In a numerical simulation we show that a modified Laplacian eigenmaps algorithm, based on the Earthmover's distance, outperforms the classic Euclidean Laplacian eigenmaps, both in terms of computational cost and the sample size needed to recover the intrinsic geometry.

Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II

We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors.

Color confinement at the boundary of the conformally compactified AdS5

The topology of closed manifolds forces interacting charges to appear in pairs. We take advantage of this property in the setting of the conformal boundary of AdS5 spacetime, topologically equivalent to the closed manifold S1× S3, by considering the coupling of two massless opposite charges on it. Taking the interaction potential as the analog of Coulomb interaction (derived from a fundamental solution of the S3 Laplace-Beltrami operator), a conformal S1× S3 metric deformation is proposed, such that free motion on the deformed metric is equivalent to motion on the round metric in the presence of the interaction potential. We give explicit expressions for the generators of the conformal algebra in the representation induced by the metric deformation.By identifying the charge as the color degree of freedom in QCD, and the two charges system as a quark-anti-quark system, we argue that the associated conformal wave operator equation could provide a realistic quantum mechanical description of the simplest QCD system, the mesons.Finally, we discuss the possibility of employing the compactification radius, R, as an- other scale along ΛQCD, by means of which, upon reparametrizing Q2c2 as (Q2c2+ħ2c2/R2), a perturbative treatment of processes in the infrared could be approached.

First- and second-order unconditionally stable direct discretization methods for multi-component Cahn–Hilliard system on surfaces

This paper proposes a first- and second-order unconditionally stable direct discretization method based on a surface mesh consisting of piecewise triangles and its dual-surface polygonal tessellation for solving the N-component Cahn–Hilliard system. We define the discretizations of the gradient, divergence, and Laplace–Beltrami operators on triangle surfaces. We prove that the proposed schemes, which combine a linearly stabilized splitting scheme, are unconditionally energy-stable. We also prove that our method satisfies the mass conservation. The proposed scheme is solved by the biconjugate gradient stabilized (BiCGSTAB) method, which can be straightforwardly applied to GPU-accelerated biconjugate gradient stabilized implementation by using the Matlab Parallel Computing Toolbox. Several numerical experiments are performed and confirm the accuracy, stability, and efficiency of our proposed algorithm.

Hodge Theory of Holomorphic Vector Bundle on Compact Kähler Hyperbolic Manifold

Abstract Let $E$ be a holomorphic vector bundle over a compact Kähler manifold $(X,\omega )$ with negative sectional curvature $sec\leq -K<0$ and $D_{E}$ be the Chern connection on $E$. In this article, we show that if $C:=|[\Lambda ,i\Theta (E)]|\leq c_{n}K$, then $(X,E)$ satisfy a family of Chern number inequalities. The main idea in our proof is to study the $L^{2}$$\bar {\partial }_{\tilde {E}}$-harmonic forms on lifting bundle $\tilde {E}$ over the universal covering space $\tilde {X}$. We also observe that there is a close relationship between the eigenvalue of the Laplace–Beltrami operator $\Delta _{\bar {\partial }_{\tilde {E}}}$ and the Euler characteristic of $X$. Precisely, if there is a line bundle $L$ on $X$ such that $\chi ^{p}(X,L^{\otimes m})$ is not constant for some integers $p\in [0,n]$, then the Euler characteristic of $X$ satisfies $(-1)^{n}\chi (X)\geq (n+1)+\lfloor \frac {c_{n}K}{2nC} \rfloor $.

Quantum Confinement for the Curvature Laplacian −Δ + cK on 2D-Almost-Riemannian Manifolds

Two-dimension almost-Riemannian structures of step 2 are natural generalizations of the Grushin plane. They are generalized Riemannian structures for which the vectors of a local orthonormal frame can become parallel. Under the 2-step assumption the singular set Z, where the structure is not Riemannian, is a 1D embedded submanifold. While approaching the singular set, all Riemannian quantities diverge. A remarkable property of these structures is that the geodesics can cross the singular set without singularities, but the heat and the solution of the Schrödinger equation (with the Laplace-Beltrami operator Δ) cannot. This is due to the fact that (under a natural compactness hypothesis), the Laplace-Beltrami operator is essentially self-adjoint on a connected component of the manifold without the singular set. In the literature such phenomenon is called quantum confinement. In this paper we study the self-adjointness of the curvature Laplacian, namely −Δ + cK, for c ∈ (0,1/2) (here K is the Gaussian curvature), which originates in coordinate-free quantization procedures (as for instance in path-integral or covariant Weyl quantization). We prove that there is no quantum confinement for this type of operators.

Tetrahedral spectral feature-Based bayesian manifold learning for grey matter morphometry: Findings from the Alzheimer's disease neuroimaging initiative.

Structural and anatomical analyses of magnetic resonance imaging (MRI) data often require a reconstruction of the three-dimensional anatomy to a statistical shape model. Our prior work demonstrated the usefulness of tetrahedral spectral features for grey matter morphometry. However, most of the current methods provide a large number of descriptive shape features, but lack an unsupervised scheme to automatically extract a concise set of features with clear biological interpretations and that also carries strong statistical power. Here we introduce a new tetrahedral spectral feature-based Bayesian manifold learning framework for effective statistical analysis of grey matter morphology. We start by solving the technical issue of generating tetrahedral meshes which preserve the details of the grey matter geometry. We then derive explicit weak-form tetrahedral discretizations of the Hamiltonian operator (HO) and the Laplace-Beltrami operator (LBO). Next, the Schrödinger's equation is solved for constructing the scale-invariant wave kernel signature (SIWKS) as the shape descriptor. By solving the heat equation and utilizing the SIWKS, we design a morphometric Gaussian process (M-GP) regression framework and an active learning strategy to select landmarks as concrete shape descriptors. We evaluate the proposed system on publicly available data from the Alzheimers Disease Neuroimaging Initiative (ADNI), using subjects structural MRI covering the range from cognitively unimpaired (CU) to full blown Alzheimer's disease (AD). Our analyses suggest that the SIWKS and M-GP compare favorably with seven other baseline algorithms to obtain grey matter morphometry-based diagnoses. Our work may inspire more tetrahedral spectral feature-based Bayesian learning research in medical image analysis.

A Data‐Driven Approach to Functional Map Construction and Bases Pursuit

AbstractWe propose a method to simultaneously compute scalar basis functions with an associated functional map for a given pair of triangle meshes. Unlike previous techniques that put emphasis on smoothness with respect to the Laplace–Beltrami operator and thus favor low‐frequency eigenfunctions, we aim for a basis that allows for better feature matching. This change of perspective introduces many degrees of freedom into the problem allowing to better exploit non‐smooth descriptors. To effectively search in this high‐dimensional space of solutions, we incorporate into our minimization state‐of‐the‐art regularizers. We solve the resulting highly non‐linear and non‐convex problem using an iterative scheme via the Alternating Direction Method of Multipliers. At each step, our optimization involves simple to solve linear or Sylvester‐type equations. In practice, our method performs well in terms of convergence, and we additionally show that it is similar to a provably convergent problem. We show the advantages of our approach by extensively testing it on multiple datasets in a few applications including shape matching, consistent quadrangulation and scalar function transfer.

Asymptotically optimal cubature formulas on manifolds for prefixed weights

Given a Riemannian manifold X with Riemannian measure μX and positive weights {ωj}j=1N, we study the conditions under which there exist points {xj}j=1N⊂X so that a cubature formula of the form (1)∫XPdμX=∑j=1NωjP(xj)holds for all polynomials P of order less than or equal to L. The problem is studied for diffusion polynomials (linear combinations of eigenfunctions of the Laplace–Beltrami operator) in the context of abstract Riemannian manifolds and for algebraic polynomials in the context of algebraic manifolds in Rn.

Functional calculus of Laplace transform type on non-doubling parabolic manifolds with ends

Let $M$ be a non-doubling parabolic manifold with ends and $L$ a non-negative self-adjoint operator on $L^{2}(M)$ which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrödinger operators $L = \Delta + V$ where $\Delta$ is the Laplace–Beltrami operator and $V$ is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of $L$ together with its time derivatives and then apply them to obtain the weak type $(1, 1)$ estimate of the functional calculus of Laplace transform type of $\sqrt{L}$ which is defined by $\mathfrak{M}(\sqrt{L}) f(x) := \int_{0}^{\infty} \bigl[\sqrt{L} e^{-t \sqrt{L}} f(x)\bigr] m(t) dt$ where $m(t)$ is a bounded function on $[0, \infty)$. In the setting of our study, both doubling condition of the measure on $M$ and the smoothness of the operators' kernels are missing. The purely imaginary power $L^{is}$, $s \in \mathbb{R}$, is a special case of our result and an example of weak type $(1, 1)$ estimates of a singular integral with non-smooth kernels on non-doubling spaces.

Computing the Riemannian curvature of image patch and single-cell RNA sequencing data manifolds using extrinsic differential geometry

Most high-dimensional datasets are thought to be inherently low-dimensional-that is, data points are constrained to lie on a low-dimensional manifold embedded in a high-dimensional ambient space. Here, we study the viability of two approaches from differential geometry to estimate the Riemannian curvature of these low-dimensional manifolds. The intrinsic approach relates curvature to the Laplace-Beltrami operator using the heat-trace expansion and is agnostic to how a manifold is embedded in a high-dimensional space. The extrinsic approach relates the ambient coordinates of a manifold's embedding to its curvature using the Second Fundamental Form and the Gauss-Codazzi equation. We found that the intrinsic approach fails to accurately estimate the curvature of even a two-dimensional constant-curvature manifold, whereas the extrinsic approach was able to handle more complex toy models, even when confounded by practical constraints like small sample sizes and measurement noise. To test the applicability of the extrinsic approach to real-world data, we computed the curvature of a well-studied manifold of image patches and recapitulated its topological classification as a Klein bottle. Lastly, we applied the extrinsic approach to study single-cell transcriptomic sequencing (scRNAseq) datasets of blood, gastrulation, and brain cells to quantify the Riemannian curvature of scRNAseq manifolds.

Bernstein inequalities via the heat semigroup

We extend the classical Bernstein inequality to a general setting including the Laplace-Beltrami operator, Schrödinger operators and divergence form elliptic operators on Riemannian manifolds or domains. We prove \(L^p\) Bernstein inqualities as well as a “reverse inequality” which is new even for compact manifolds (with or without boundary). Such a reverse inequality can be seen as the dual of the Bernstein inequality. The heat kernel will be the backbone of our approach but we also develop new techniques. For example, once reformulating Bernstein inequalities in a semi-classical fashion we prove and use weak factorization of smooth functions à la Dixmier–Malliavin and BMO–\(L^\infty \) multiplier results (in contrast to the usual \(L^\infty \)–BMO ones). Also, our approach reveals a link between the \(L^p\)-Bernstein inequality and the boundedness on \(L^p\) of the Riesz transform.

Diameter and Laplace Eigenvalue Estimates for Left-invariant Metrics on Compact Lie Groups

Let $G$ be a compact connected Lie group of dimension $m$. Once a bi-invariant metric on $G$ is fixed, left-invariant metrics on $G$ are in correspondence with $m\times m$ positive definite symmetric matrices. We estimate the diameter and the smallest positive eigenvalue of the Laplace-Beltrami operator associated to a left-invariant metric on $G$ in terms of the eigenvalues of the corresponding positive definite symmetric matrix. As a consequence, we give partial answers to a conjecture by Eldredge, Gordina and Saloff-Coste; namely, we give large subsets $\mathcal S$ of the space of left-invariant metrics $\mathcal M$ on $G$ such that there exists a positive real number $C$ depending on $G$ and $\mathcal S$ such that $\lambda_1(G,g)\operatorname{diam}(G,g)^2\leq C$ for all $g\in\mathcal S$. The existence of the constant $C$ for $\mathcal S=\mathcal M$ is the original conjecture.

Spectra, rigidity and stability of sine-cones

We compute the spectra of the Laplace-Beltrami operator, the connection Laplacian on 1-forms and the Einstein operator on symmetric 2-tensors on the sine-cone over a positive Einstein manifold (M,g). We conclude under which conditions on (M,g), the sine-cone is dynamically stable under the singular Ricci-de Turck flow and rigid as a singular Einstein manifold.

Theorems of Chernoff and Ingham for certain eigenfunction expansions

We prove an uncertainty principle for certain eigenfunction expansions on L2(R+,w(r)dr) and use it to prove analogues of theorems of Chernoff and Ingham for Laplace-Beltrami operators on compact symmetric spaces, special Hermite operator on Cn and Hermite operator on Rn.

Rotational Hypersurfaces Satisfying ∆^I R=AR in the Four-Dimensional Euclidean Space

In this study, rotational hypersurfaces in the 4-dimensional Euclidean space are discussed. Some relations of curvatures of hypersurfaces are given, such as the mean, Gaussian, and their minimality and flatness. In addition, Laplace-Beltrami operator has been defined for 4-dimensional hypersurfaces depending on the first fundamental form. Moreover, it is shown that each element of the 4×4 order matrix A, which satisfies the condition ∆^I R=AR, is zero, that is, the rotational hypersurface R is minimal.