Spectral properties of symmetrized AMV operators

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 uniformly locally doubling 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.