A lower estimate for first Neumann eigenvalue of compact manifolds (Q2837913)

Let \(\lambda_1\) be the first Neumann eigenvalue of a compact Riemannian manifold \(M\) of dimension \(m\). The authors assume that the Ricci curvature satisfies the estimate \(\mathrm{Ric}(M)\geq-(n-1)K\) and that the boundary of \(M\) is convex, i.e. the second fundamental form with respect to the outward normal vector field is positive definite. Let \(d\) be the diameter of \(M\).NEWLINENEWLINENEWLINEThe authors show NEWLINE\[NEWLINE\lambda_1\geq{1\over{2(n-1)d^2}}\exp\{-1-\sqrt{1+2(n-1)^2d^2K}\}\,.NEWLINE\]NEWLINE This improves earlier estimates due to \textit{S.-T. Yau} [Ann. Sci. Éc. Norm. Supér. (4) 8, 487--507 (1975; Zbl 0325.53039)] and \textit{P. Li} and \textit{S.-T. Yau} [Proc. Symp. Pure Math., Vol. 36, 205--239 (1980; Zbl 0441.58014)].