Noncommutative Laplacian and numerical approximation of Laplace-Beltrami spectrum of compact Riemann surfaces

Abstract

We derive a numerical approximation of the Laplace-Beltrami operator on compact surfaces embedded in ${\mathbb{R}}^3$ with an axial symmetry. To do so we use a noncommutative Laplace operator defined on the space of finite dimensional hermitian matrices. This operator is derived from a foliation of the surface obtained under an $S^1$-action on the surface. We present numerical results in the case of the sphere and a generic ellipsoid.