An exact solution to an equation and the first eigenvalue of a compact manifold
Let \((M,g)\) be a closed Riemannian manifold of dimension \(n\) and let \(\lambda\) be the first non-zero eigenvalue of the scalar Laplacian. Assume that the Ricci curvature has a negative lower bound \((n-1)\kappa<0\). Let \(r\) be the largest interior radius of the nodal domains of the first eigenvalue. Let \(d=2r\). The author establishes the estimates: \(\lambda\geq{{\pi^2}\over{d^2[1-(n-1)\kappa/(2\lambda)]}}>0\) and \(\lambda\geq{\pi^2\over d^2}+{1\over2}(n-1)\kappa\,.\)
scientific article; zbMATH DE number 5316795