Spectral properties of symmetrized AMV operators

TL;DR

This paper investigates the spectral properties of the symmetrized Asymptotic Mean Value Laplacian on metric measure spaces, establishing eigenvalue existence and convergence to classical Laplacians on Riemannian manifolds.

Contribution

It introduces spectral analysis of the symmetrized AMV operators and proves their convergence to Laplace--Beltrami operators on manifolds.

Findings

Existence of isolated eigenvalues for the operators $ ilde{ riangle}_r$

Spectral convergence of $ ilde{ riangle}_r$ to classical Laplacians

Eigenvalues characterized via min-max procedures

Abstract

The symmetrized Asymptotic Mean Value Laplacian $\tilde{\Delta}$, obtained as limit of approximating operators $\tilde{\Delta}_r$, is an extension of the classical Euclidean Laplace operator to the realm of metric measure spaces. We show that, as $r \downarrow 0$, the operators $\tilde{\Delta}_r$ eventually admit isolated eigenvalues defined via min-max procedure on any compact locally Ahlfors regular metric measure space. Then we prove $L^2$ and spectral convergence of $\tilde{\Delta}_r$ to the Laplace--Beltrami operator of a compact Riemannian manifold, imposing Neumann conditions when the manifold has a non-empty boundary.