Approximating sub-riemannian structures by Riemannian metrics and spectral convergence

TL;DR

This paper develops a method to approximate sub-Riemannian structures using Riemannian metrics, demonstrating convergence of volumes and spectral properties of Laplace operators, advancing understanding of geometric analysis in sub-Riemannian geometry.

Contribution

It introduces a novel sequence of Riemannian metrics approximating sub-Riemannian structures and proves convergence of volumes and spectral properties, bridging the gap between Riemannian and sub-Riemannian analysis.

Findings

Riemannian volumes converge to Popp's volume

Spectral convergence of Laplace operators established

Provides a new approach for analyzing sub-Riemannian geometries

Abstract

We construct an approximating sequence of Riemannian metrics tailored to a given sub-Riemannian structure. We prove that the sequence of associated Riemannian volumes converge to Popp's volume and we then proceed to study the spectral convergence of the associated Laplace operators.