Closure of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds and semi-Fredholm properties of differential operators on Lie manifolds

TL;DR

This paper investigates the domain closure of the Laplace-Beltrami operator on 2D almost-Riemannian manifolds, introducing a method that handles complex singularities at tangency points using Lie groupoid tools.

Contribution

It presents a novel approach to determine the operator's domain on manifolds with tangency points, a challenging class of singular geometries.

Findings

Established natural domains for Laplace-Beltrami perturbations

Extended analysis to geometries with tangency points

Provided semi-Fredholm properties of differential operators

Abstract

The problem of determining the domain of the closure of the Laplace-Beltrami operator on a 2D almost-Riemannian manifold is considered. Using tools from theory of Lie groupoids natural domains of perturbations of the Laplace-Beltrami operator are found. The main novelty is that the presented method allows us to treat geometries with tangency points. This kind of singularity is difficult to treat since those points do not have a tubular neighbourhood compatible with the almost-Riemannian metric.