The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

2005/06/13 by Daniel Grieser, Grieser, Daniel
Computer Science · Medicine · #35P15 #51M16 #Advanced Mathematical Modeling in Engineering #Advanced Neuroimaging Techniques and Applications #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP) #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.math/0506243

openalex publication_date 2005/06/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show how 'test' vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger's constant may be obtained precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger's constant in terms of the inradius of a plane domain.

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