The paper starts with a discussion involving the Sobolev constant on geodesic balls and then follows with a derivation of a lower bound for the first eigenvalue of the Laplacian on manifolds with small negative curvature. The derivation involves Moser iteration.
"A Global Curvature Pinching Result of the First Eigenvalue of the Laplacian on Riemannian Manifolds." Abstr. Appl. Anal. 2013 1 - 5, 2013. https://doi.org/10.1155/2013/237418