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This paper investigates links between the eigenvalues and eigenfunctions ofthe Laplace-Beltrami operator, and the higher Cheeger constants of smoothRiemannian manifolds, possibly weighted and/or with boundary. The higherCheeger constants give a loose description of the major geometric features of amanifold. We give a constructive upper bound on the higher Cheeger constants,in terms of the eigenvalue of any eigenfunction with the corresponding numberof nodal domains. Specifically, we show that for each such eigenfunction, apositive-measure collection of its superlevel sets have their Cheeger ratiosbounded above in terms of the corresponding eigenvalue.  Some manifolds have their major features entwined across severaleigenfunctions, and no single eigenfunction contains all the major features. Inthis case, there may exist carefully chosen linear combinations of theeigenfunctions, each with large values on a single feature, and small valueselsewhere. We can then apply a soft-thresholding operator to these linearcombinations to obtain new functions, each supported on a single feature. Weshow that the Cheeger ratios of the level sets of these functions also give anupper bound on the Laplace-Beltrami eigenvalues. We extend these level setresults to nonautonomous dynamical systems, and show that the dynamic Laplacianeigenfunctions reveal sets with small dynamic Cheeger ratios.

Higher Cheeger ratios of features in Laplace-Beltrami eigenfunctions