Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions

TL;DR

This paper explores the relationship between Laplace-Beltrami eigenfunctions and higher Cheeger constants, providing bounds and methods to identify major geometric features of manifolds through eigenfunction analysis.

Contribution

It introduces a constructive upper bound on higher Cheeger constants using eigenfunctions and develops techniques to extract features via linear combinations and level sets.

Findings

Upper bounds on Cheeger constants via eigenvalues and eigenfunctions

Methods to isolate manifold features using linear combinations of eigenfunctions

Extension of results to dynamic systems and dynamic Cheeger ratios

Abstract

This paper investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly weighted and/or with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We give a constructive upper bound on the higher Cheeger constants, in terms of the eigenvalue of any eigenfunction with the corresponding number of nodal domains. Specifically, we show that for each such eigenfunction, a positive-measure collection of its superlevel sets have their Cheeger ratios bounded above in terms of the corresponding eigenvalue. Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios.