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

TL;DR

The paper tackles computing the Laplace–Beltrami spectrum on compact Riemann surfaces embedded in $\mathbb{R}^3$ with $S^1$ symmetry by replacing the commutative function algebra with finite-dimensional Hermitian matrix algebras via matrix regularization. It defines a noncommutative Laplacian built from commutators of quantized coordinates and proves a convergence theorem: eigenmatrices of the noncommutative Laplacian converge to eigenfunctions of $\Delta_g$ with eigenvalues converging as $\hbar\to0$, for toric surfaces. Numerically, it demonstrates spectra for the sphere, ellipsoid, standard torus, and a genus-2 surface, showing convergence and robustness across geometries. The approach offers a structure-preserving, mesh-free discretization applicable to higher-genus surfaces and has potential implications for spectral geometry, graphics, and mathematical physics.

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.

Figures (4)

• Figure 1: Foliation of embedded compact surfaces in $\mathbb{R}^3$
• Figure 2: Quantized coordinates for the embedded torus $\mathbb{T}^2$
• Figure 3: Slices of the embedded torus for different values of $z$.
• Figure 4: First set of eigenvalues of the Laplace-Beltrami operator of the embedded double torus.

Theorems & Definitions (29)

• Theorem 1.1
• Definition 2.1: Toric manifold da_silva_lectures_2008
• Example 2.1
• Lemma 3.1
• proof
• Definition 3.1: Matrix quantization
• Example 3.1: The unit sphere
• Theorem 3.1
• proof
• Definition 3.2: Smooth matrix quantization