  Sharp comparison theorems are derived for all eigenvalues of the (weighted)
Laplacian, for various classes of weighted-manifolds (i.e. Riemannian manifolds
endowed with a smooth positive density). Examples include Euclidean space
endowed with strongly log-concave and log-convex densities, extensions to
$p$-exponential measures, unit-balls of $\ell_p^n$, one-dimensional spaces and
Riemannian submersions. Our main tool is a general Contraction Principle for
"eigenvalues" on arbitrary metric-measure spaces. Motivated by Caffarelli's
Contraction Theorem, we put forth several conjectures pertaining to the
existence of contractions from the canonical sphere (and Gaussian space) to
weighted-manifolds of appropriate topological type having (generalized) Ricci
curvature positively bounded below; these conjectures are consistent with all
known isoperimetric, heat-kernel and Sobolev-type properties, and would imply
sharp conjectural spectral estimates on such spaces. While we do not resolve
these conjectures for the individual eigenvalues, we verify their Weyl
asymptotic distribution in the compact and non-compact settings, obtain
non-asymptotic estimates using the Cwikel--Lieb--Rozenblum inequality, and
estimate the trace of the associated heat-kernel assuming that the associated
heat semi-group is hypercontractive. As a side note, an interesting trichotomy
for the heat-kernel is obtained.
