First eigenvalue for the \(p\)-Laplace operator (Q1968745)

This paper deals with the first eigenvalue of the \(p\)-Laplacian operator \(\Delta_p\), \(p\geq 1\), on \(m\)-dimensional Riemannian manifolds. The first result is a generalization of a comparison theorem proved by \textit{S. Y. Cheng} [Math. Z. 143, 289-297 (1975; Zbl 0329.53035)] for \(p=2\). The next result is a Faber-Krahn-type isoperimetric inequality for the \(p\)-Laplacian in bounded domains of a manifold with positive Ricci curvature, which was obtained in the case of the Laplacian by \textit{P. Bérard}, \textit{D. Meyer} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 15, 513-541 (1982; Zbl 0527.35020)]. A comparison result of Lichnerowicz-Obata and Cheeger-type estimates are also generalized to the case \(p>1\).