About: Cheng's eigenvalue comparison theorem

In Riemannian geometry, Cheng's eigenvalue comparison theorem states in general terms that when a domain is large, the first Dirichlet eigenvalue of its Laplace–Beltrami operator is small. This general characterization is not precise, in part because the notion of "size" of the domain must also account for its curvature. The theorem is due to by Shiu-Yuen Cheng. Using geodesic balls, it can be generalized to certain tubular domains.