A generalization of Cheng's theorem (Q1018242)

In his paper [``Eigenvalue comparison theorems and its geometric application'', Math. Z. 143, 289--297 (1975; Zbl 0329.53035)], \textit{S. Y. Cheng} proved a comparison theorem for the first eigenvalue of a geodesic ball. By taking the radius of the ball to infinity, he obtained an estimate for the bottom of the \(L^2\) spectrum. In this paper, the authors prove a generalization of the theorem of Cheng on the upper bound of the bottom of the \(L^2\) spectrum for a complete Riemann manifold.