An exact solution to an equation and the first eigenvalue of a compact manifold

Abstract

We study an exact solution to a singular ordinary differential equation and use the solution to give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with a negative lower bound on the Ricci curvature in terms of the lower bound on the Ricci curvature and the largest interior radius of the nodal domains of the eigenfunction. This provides a new way to estimate eigenvalues.