  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.
