The cotan formula constitutes a discretization of the Laplace-Beltrami operator on polyhedral surfaces in a Finite Element sense. In this note we give an overview over its convergence properties. The mean curvature vector, given by the Laplacian of the embedding of a surface, will serve as a case study: It will be shown that mean curvature viewed as a functional converges, whereas the correspon...

We consider restrictions along closed geodesics and geodesic circles for eigenfunctions of the Laplace-Beltrami operator on a compact hyperbolic Riemann surface. We obtain a non-trivial bound on the L-norm of such restrictions as the eigenvalue tends to infinity. We use methods from the theory of automorphic functions and in particular the uniqueness of invariant functionals on irreducible unit...

Persistent homology provides a new approach for the topological simplification of big data via measuring the life time of intrinsic topological features in a filtration process and has found its success in scientific and engineering applications. However, such a success is essentially limited to qualitative data characterization, identification and analysis (CIA). Indeed, persistent homology ha...

The notion of Lp–distributions is introduced on Riemannian symmetric spaces of noncompact type and their main properties are established. We use a geometric description for the topology of the space of test functions in terms of the Laplace–Beltrami operator. The techniques are based on a-priori estimates for elliptic operators. We show that structure theorems, similar to Rn, hold on symmetric ...

The cotangent formula constitutes an intrinsic discretization of the Laplace– Beltrami operator on polyhedral surfaces in a finite element sense. This note gives an overview of approximation and convergence properties of discrete Laplacians and mean curvature vectors for polyhedral surfaces located in the vicinity of a smooth surface in euclidean 3–space. In particular, we show that mean curvat...

We consider the Klein–Gordon equation associated with the Laplace– Beltrami operator ∆ on real hyperbolic spaces of dimension n≥2; as ∆ has a spectral gap, the wave equation is a particular case of our study. After a careful kernel analysis, we obtain dispersive and Strichartz estimates for a large family of admissible couples. As an application, we prove global well–posedness results for the c...

We derive error estimates for the piecewise linear finite element approximation of the Laplace–Beltrami operator on a bounded, orientable, C3, surface without boundary on general shape regular meshes. As an application, we consider a problem where the domain is split into two regions: one which has relatively high curvature and one that has low curvature. Using a graded mesh we prove error esti...

We consider \geometric" scattering for a Laplace-Beltrami operator on a compact Riemannian manifold inserted between \wires," that is, two half-lines. We discuss applicability and correctness of this model. With an example, we show that such a scattering problem may exhibit unusual properties: the transition coeecient has a sequence of sharp peaks which become more and more distant at high ener...

We consider “geometric” scattering for a Laplace-Beltrami operator on a compact Riemannian manifold inserted between “wires,” that is, two half-lines. We discuss applicability and correctness of this model. With an example, we show that such a scattering problem may exhibit unusual properties: the transition coefficient has a sequence of sharp peaks which become more and more distant at high en...

Ω 7−→ λ1(Ω) , under the volume constraint Vol(Ω) = α (where α ∈ (0,Vol(M)) is fixed) are called extremal domains. Smooth extremal domains are characterized by the property that the eigenfunctions associated to the first eigenvalue of the Laplace-Beltrami operator have constant Neumann boundary data [2]. In other words, a smooth domain is extremal if and only if there exists a positive function ...