  We use the averaged variational principle introduced in a recent article on
graph spectra [7] to obtain upper bounds for sums of eigenvalues of several
partial differential operators of interest in geometric analysis, which are
analogues of Kr{\"o}ger 's bound for Neumann spectra of Laplacians on Euclidean
domains [12]. Among the operators we consider are the Laplace-Beltrami operator
on compact subdomains of manifolds. These estimates become more explicit and
asymptotically sharp when the manifold is conformal to homogeneous spaces (here
extending a result of Strichartz [21] with a simplified proof). In addition we
obtain results for the Witten Laplacian on the same sorts of domains and for
Schr{\"o}dinger operators with confining potentials on infinite Euclidean
domains. Our bounds have the sharp asymptotic form expected from the Weyl law
or classical phase-space analysis. Similarly sharp bounds for the trace of the
heat kernel follow as corollaries.
