Some upper bounds for sums of eigenvalues of the Neumann Laplacian

Let $\mu_{k}(\Omega)$ be the $k$th Neumann eigenvalue of a bounded domain $\Omega$ with piecewisely smooth boundary in $\textbf{R}^{n}$. In 1992, P. Kröger proved that $k^{-\frac{n+2}{n}}\sum_{j=1}^{k}\mu_{j}\leq{4n\pi^{2}\over n+2}( \omega_{n}V)^{-2/n}$, where the upper bound is sharp in view of Weyl's asymptotic formula. The aim of this paper is twofold. First, we will improve this estimate by multiplying a factor in terms of $k$ to its right-hand side which approaches strictly from below to 1 as $k$ tends to infinity. Second, we will generalize Kröger's estimate to the case when $\Omega$ is a compact Euclidean submanifold.