Eigenvalues estimate for the Neumann problem of a bounded domain
Let \((M,g)\) be a complete \(n\)-dimensional Riemannian manifold with Ricci curvature bounded below by \(-(n-1)a^2\) for some \(a\geq0\). Let \(\Omega\) be a bounded domain in \((M,g)\) with smooth boundary of volume \(V(\Omega)\). Denote the eigenvalues of the Neumann Laplacian on \(\Omega\) by \(0=\lambda_1(\Omega)\leq\dots \leq\Omega_k(\Omega)\leq\dots\). The authors show: Theorem: There exist constants \(A_n>0\) and \(B_n>0\) so that \(\lambda_k(\Omega)\leq A_na^2+B_n({k\over V(\Omega)})^{2/n}\).
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