On the Cheeger inequality in Carnot-Carathéodory spaces

2023/12/20 by Martijn Kluitenberg, Kluitenberg, Martijn
Computer Science · Mathematics · Medicine · #35P15 #58C40 #58J50 #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #Nonlinear Partial Differential Equations #Orthopaedic implants and arthroplasty #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2312.13058

openalex publication_date 2023/12/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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.

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