Hyperbolic Eisenstein series on n-dimensional hyperbolic spaces

We define a generalized hyperbolic Eisenstein series for a pair of a hyperbolic manifold of finite volume and its submanifold. We prove the convergence, the differential equation and the precise spectral expansion associated to the Laplace–Beltrami operator. We also derive the analytic continuation with the location of the possible poles and their residues from the spectral expansion.     

On the eigenfunction expansion of the Laplace–Beltrami operator in hyperbolic space

We describe the spectral projection of the Laplace–Beltrami operator in n-dimensional hyperbolic space by studying its resolvent as an analytic operator-valued function and applying the technique of contour integration. As a result an integral formula is established for the associated Legendre function     

On the approximation of some special functions in Ramanujan’s generalized modular equation with signature 3 $$^{*}$$ ∗

The Ramanujan Journal - In this work we investigate the heat kernel of the Laplace–Beltrami operator on a rectangular torus and the according temperature distribution. We compute the minimum...     

Pseudo‐Spectral Methods for the Laplace‐Beltrami Equation and the Hodge Decomposition on Surfaces of Genus One

The inversion of the Laplace‐Beltrami operator and the computation of the Hodge decomposition of a tangential vector field on smooth surfaces arise as computational tasks in many areas of science, from computer graphics to machine learning to computational physics. Here, we present a high‐order accurate pseudo‐spectral approach, applicable to closed surfaces of genus one in three‐dimensional space, with a view toward applications in plasma physics and fluid dynamics. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 941–955, 2017     

The wave equation on hyperbolic spaces

We study the dispersive properties of the wave equation associated with the shifted Laplace–Beltrami operator on real hyperbolic spaces and deduce new Strichartz estimates for a large family of admissible pairs. As an application, we obtain local well-posedness results for the nonlinear wave equation.     

The p-widths of a surface

Publications mathématiques de l'IHÉS - The $p$ -widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which...     

Absence of eigenvalue at the bottom of the continuous spectrum on asymptotically hyperbolic manifolds

For a class of asymptotically hyperbolic manifolds, we show that the bottom of the continuous spectrum of the Laplace–Beltrami operator is not an eigenvalue. Our approach only uses properties of the operator near infinity and, in particular, does not require any global assumptions on the topology or the curvature, unlike previous papers on the same topic.     

Closed form formulae for the heat kernels and the Green functions for the Laplacians on the symmetric spaces of rank one

We show some integral representations of the heat kernels and explicit expressions of the Green functions for the Laplace–Beltrami operators on three series of hyperbolic spaces.     

Extremal eigenvalues of the Laplacian on Euclidean domains and closed surfaces

We investigate properties of the sequences of extremal values that could be achieved by the eigenvalues of the Laplacian on Euclidean domains of unit volume, under Dirichlet and Neumann boundary conditions, respectively. In a second part, we study sequences of extremal eigenvalues of the Laplace–Beltrami operator on closed surfaces of unit area.     

Fractional Cauchy problems on compact manifolds

We investigate anomalous diffusion on compact Riemannian manifolds, modeled by time-changed Brownian motions. These stochastic processes are governed by equations involving the Laplace–Beltrami operator and a time-fractional derivative of order β ∈ (0, 1). We also consider time dependent random fields that can be viewed as random fields on randomly varying manifolds.     

Invariants of Isospectral Deformations and Spectral Rigidity

We introduce a notion of weak isospectrality for continuous deformations. Consider the Laplace–Beltrami operator on a compact Riemannian manifold with Robin boundary conditions. Given a Kronecker invariant torus Λ of the billiard ball map with a Diophantine vector of rotation we prove that certain integrals on Λ involving the function in the Robin boundary conditions remain constant under weak isospectral deformations. To this end we construct continuous families of quasimodes associated with Λ. We obtain also isospectral invariants of the Laplacian with a real-valued potential on a compact manifold for continuous deformations of the potential. These invariants are obtained from the first Birkhoff invariant of the microlocal monodromy operator associated to Λ. As an application we prove spectral rigidity of the Robin boundary conditions in the case of Liouville billiard tables of dimension two in the presence of a (?/2?)² group of symmetries.     

A remark on the Gaussian lower bound for the Neumann heat kernel of the Laplace–Beltrami operator

We adapt in the present note the perturbation method introduced in Choulli and Kayser (Positivity 19(3):625–646, 2015) to get a lower Gaussian bound for the Neumann heat kernel of the Laplace–Beltrami operator on an open subset of a compact Riemannian manifold.     

Problems for equations with special parabolic operator of fractional differentiation

We establish the well-posedness of the Cauchy problem and the two-point boundary-value problem for an equation with an operator of fractional differentiation that corresponds to the singular parabolic Beltrami – Laplace operator on a surface of the Dini class.     

A spectral notion of Gromov–Wasserstein distance and related methods

We introduce a spectral notion of distance between objects and study its theoretical properties. Our distance satisfies the properties of a metric on the class of isometric shapes, which means, in particular, that two shapes are at 0 distance if and only if they are isometric when endowed with geodesic distances. Our construction is similar to the Gromov–Wasserstein distance, but rather than viewing shapes merely as metric spaces, we define our distance via the comparison of heat kernels. This allows us to establish precise relationships of our distance to previously proposed spectral invariants used for data analysis and shape comparison, such as the spectrum of the Laplace–Beltrami operator, the diagonal of the heat kernel, and certain constructions based on diffusion distances. In addition, the heat kernel encodes a natural notion of scale, which is useful for multi-scale shape comparison. We prove a hierarchy of lower bounds for our distance, which provide increasing discriminative power at the cost of an increase in computational complexity. We also explore the definition of other spectral metrics on collections of shapes and study their theoretical properties.     

A Laplace Operator on Semi-Discrete Surfaces

This paper studies a Laplace operator on semi-discrete surfaces. A semi-discrete surface is represented by a mapping into three-dimensional Euclidean space possessing one discrete variable and one continuous variable. It can be seen as a limit case of a quadrilateral mesh, or as a semi-discretization of a smooth surface. Laplace operators on both smooth and discrete surfaces have been an object of interest for a long time, also from the viewpoint of applications. There are a wealth of geometric objects available immediately once a Laplacian is defined, e.g., the mean curvature normal. We define our semi-discrete Laplace operator to be the limit of a discrete Laplacian on a quadrilateral mesh, which converges to the semi-discrete surface. The main result of this paper is that this limit exists under very mild regularity assumptions. Moreover, we show that the semi-discrete Laplace operator inherits several important properties from its discrete counterpart, like symmetry, positive semi-definiteness, and linear precision. We also prove consistency of the semi-discrete Laplacian, meaning that it converges pointwise to the Laplace–Beltrami operator, when the semi-discrete surface converges to a smooth one. This result particularly implies consistency of the corresponding discrete scheme.     

Eigenvalues under the Backward Ricci Flow on Locally Homogeneous Closed 3-manifolds

In this paper, we study the evolving behaviors of the first eigenvalue of the 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 sub-Riemannian geometry after a proper rescaling, the eigenvalue evolves toward zero.     

$$L^p$$ Joint Eigenfunction Bounds on Quaternionic Spheres

We prove some sharp \(L^p-L^2\) estimates for joint spectral projections \(\pi _{\ell \ell '}\), with \(\ell ,\ell '\in {\mathbb {N}}\), \(\ell \ge \ell '\ge 0\), \(1\le p\le 2\), associated to the Laplace–Beltrami operator and to a suitably defined subLaplacian on the unit quaternionic sphere.     

Lagrange multiplier and singular limit of double obstacle problems for the Allen–Cahn equation with constraint

We study the properties of the Lagrange multiplier for an Allen–Cahn equation with a double obstacle potential. Here, the dynamic boundary condition, including the Laplace–Beltrami operator on the boundary, is investigated. We then establish the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier of our problem. We present remarks on a trace problem as well as on the Neumann boundary condition. Moreover, we describe a numerical experiment for a problem with Neumann boundary condition using the Lagrange multiplier. Copyright © 2016 John Wiley & Sons, Ltd.     

A Theorem of DoCarmo–Zhou Type: Oscillations and Estimates for the First Eigenvalue of the p-Laplacian

It is shown that the estimates, obtained by DoCarmo and Zhou (Trans. Amer. Math. Soc. 351(4):1391–1401, 1999) for the first eigenvalue of the Laplace–Beltrami operator on open manifolds via an oscillation theorem, can be naturally extended to the semi-elliptic singular operator, the p-Laplacian on manifolds, defined by ${\Delta_p(u) := -div (||\nabla u||^{p-2}\nabla u)}$ .     

Elliptic differential inclusions on non-compact Riemannian manifolds

We investigate a large class of elliptic differential inclusions on non-compact complete Riemannian manifolds which involves the Laplace–Beltrami operator and a Hardy-type singular term. Depending on the behavior of the nonlinear term and on the curvature of the Riemannian manifold, we guarantee non-existence and existence/multiplicity of solutions for the studied differential inclusion. The proofs are based on nonsmooth variational analysis as well as isometric actions and fine eigenvalue properties on Riemannian manifolds. The results are also new in the smooth setting.