Let M denote a complete, simply connected Riemannian manifold with sectional curvature K M ≤ k and Ricci curvature Ric M ≥ (n − 1)K, where k, K ∈ R. Then for a bounded domain Ω ⊂ M with smooth boundary, we prove that the first nonzero Neumann eigenvalue µ 1 (Ω) ≤ Cµ 1 (B k (R)). Here B k (R) is a geodesic ball of radius R > 0 in the simply connected space form M k such that vol(Ω) = vol(B k (R)), and C is a constant which depends on the volume, diameter of Ω and the dimension of M.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.