Spectral properties of symmetrized AMV operators

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.