Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature
Let \(M\) be a compact \(n\) dimensional Riemannian manifold with Ricci tensor \(\rho\) and diameter \(d\). The boundary, if any, is said to be weakly convex if the second fundamental form with respect to the outward normal is nonnegative; let \(\kappa\) be the associated mean curvature and let \(r\) be the inscribed radius. Impose suitable boundary conditions and let \(\lambda\) be the first positive eigenvalue of the scalar Laplacian. \textit{A. Lichnérowicz} [`Géométrie des groupes de transformations' (1984; Zbl 0096.16001)] showed that if \(M\) is closed and if \(\rho\geq(n-1)K>0\), then \(\lambda\geq nK\). The author generalizes work of \textit{P. Li} and \textit{S.-T. Yau} [Proc. Symp. Pure Math. 36, 205-239 (1980; Zbl 0441.58014)] and also of \textit{H. Yang} and \textit{J. Zhong} [Sci. Sin., Ser. A 27, 1265-1273 (1984; Zbl 0561.53046)] to establish two related results: Theorem 1. If \(M\) is closed and if \(\rho\geq(n-1)K\geq 0\), then \(\lambda\geq{1\over 4}(n-1)K+{{\pi^2}\over{d^2}}\). Theorem 2. Assume that \(\partial M\) is non-empty and that \(\rho\geq(n-1)K\geq 0\). a) If \(\partial M\) is weakly convex, then the first positive Neumann eigenvalue \(\lambda\) of \(M\) satisfies \(\lambda\geq{1\over 4}(n-1)K+{{\pi^2}\over{d^2}}\). b) If \(\kappa\geq 0\), then the first positive Dirichlet eigenvalue \(\lambda\) satisfies \(\lambda\geq{1\over 4}\{(n-1)K+{{\pi^2}\over {r^2}}\}\).
scientific article; zbMATH DE number 1463708