# On the Cheeger Inequality in Carnot-Carathéodory Spaces

## TL;DR

This paper extends a mathematical inequality to certain geometric spaces and provides a method to estimate a key constant.

## Contribution

The paper generalizes the Cheeger inequality and proves a version of Courant’s theorem for mixed boundary conditions in Carnot-Carathéodory spaces.

## Key findings

- A Cheeger inequality is generalized to rank-varying Carnot-Carathéodory spaces.
- A new method is introduced to lower bound the Cheeger constant.
- A version of Courant’s nodal domain theorem is proven for Neumann and mixed boundary conditions.

## Abstract

We generalize the Cheeger inequality, a lower bound on the first nontrivial eigenvalue of a Laplacian, to the case of geometric sub-Laplacians on rank-varying Carnot-Carathéodory spaces and we describe a concrete method to lower bound the Cheeger constant. The proof is geometric, and works for Dirichlet, Neumann and mixed boundary conditions. One of the main technical tools in the proof is a generalization of Courant’s nodal domain theorem, which is proven from scratch for Neumann and mixed boundary conditions. Carnot groups and the Baouendi-Grushin cylinder are treated as examples.