On the Cheeger inequality in Carnot-Carath\'eodory spaces
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-Carath\'eodory 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.
Keywords
Cite
@article{arxiv.2312.13058,
title = {On the Cheeger inequality in Carnot-Carath\'eodory spaces},
author = {Martijn Kluitenberg},
journal= {arXiv preprint arXiv:2312.13058},
year = {2025}
}
Comments
Updates v2: - Added a proof of Courant's theorem which does not use (S), improving the main result - Added Section 6 on methods for obtaining lower bounds for the Cheeger constant - Minor readability improvements Updates v3: Fixed some typos. Updates v4: Final published version