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On the Cheeger inequality in Carnot-Carathéodory spaces

Martijn Kluitenberg

TL;DR

This work extends the Cheeger inequality to geometric sub-Laplacians on Carnot-Carathéodory spaces, including rank-varying structures, and develops a fully geometric route to bound the Cheeger constant under Dirichlet, Neumann, and mixed boundary conditions. A central contribution is the Neumann-Ccheeger bound λ_2^N(Ω) ≥ (1/4) h_N(Ω)^2, supported by a generalized Courant nodal-domain theorem tailored to CC-spaces and both proofs that relax assumptions while preserving validity. The authors introduce a max-flow/min-cut framework to systematically obtain lower bounds on h_N and h_D via horizontal vector fields, and they demonstrate the theory on concrete models such as Carnot groups and the Grushin cylinder, including explicit constants and spectra where available. Overall, the paper broadens spectral geometry in sub-Riemannian settings, providing practical tools for estimating eigenvalues through geometric quantities in CC-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-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.

On the Cheeger inequality in Carnot-Carathéodory spaces

TL;DR

This work extends the Cheeger inequality to geometric sub-Laplacians on Carnot-Carathéodory spaces, including rank-varying structures, and develops a fully geometric route to bound the Cheeger constant under Dirichlet, Neumann, and mixed boundary conditions. A central contribution is the Neumann-Ccheeger bound λ_2^N(Ω) ≥ (1/4) h_N(Ω)^2, supported by a generalized Courant nodal-domain theorem tailored to CC-spaces and both proofs that relax assumptions while preserving validity. The authors introduce a max-flow/min-cut framework to systematically obtain lower bounds on h_N and h_D via horizontal vector fields, and they demonstrate the theory on concrete models such as Carnot groups and the Grushin cylinder, including explicit constants and spectra where available. Overall, the paper broadens spectral geometry in sub-Riemannian settings, providing practical tools for estimating eigenvalues through geometric quantities in CC-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-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.
Paper Structure (16 sections, 30 theorems, 109 equations, 1 figure, 1 table)

This paper contains 16 sections, 30 theorems, 109 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Carnot-Carathéodory spaces
  4. Sub-Laplacians
  5. Quadratic form and extension
  6. Boundary conditions
  7. Coarea formula
  8. Courant's nodal domain theorem
  9. First proof
  10. Second proof
  11. Dirichlet-Cheeger inequality
  12. Neumann-Cheeger inequality
  13. Max flow min cut result
  14. Examples
  15. Carnot groups
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $M$ be a CC-space and $\Omega \subseteq M$ be a connected bounded domain with piecewise smooth boundary. Assume that Then, we have where where the infimum is taken over all smooth (not necessarily connected) hypersurfaces $\Sigma \subseteq \Omega$ that separate $\Omega$ into two disjoint open sets $\Omega_1$ and $\Omega_2$. Here, $\omega$ is a smooth measure on $M$ and $\sigma$ is the induc

Figures (1)

  • Figure 1: A domain $\Omega \subseteq M$ with nodal set $Z_u$ in red and Dirichlet boundary conditions enforced on $\Gamma \subseteq \partial \Omega$ in orange. One of the nodal domains is separated from the boundary of $\Omega$, while the other two touch it.

Theorems & Definitions (64)

  • Theorem 1.1: Neumann-Cheeger inequality for CC-spaces
  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Remark 2.5
  • Proposition 2.6: Sub-Riemannian Gauss-Green formula
  • Proposition 2.7
  • proof
  • Lemma 2.8: Vanishing Lemma
  • ...and 54 more