Manifolds With Bounded Integral Curvature and no Positive Eigenvalue Lower Bounds - Anderson, Connor; Ramos Olivé, Xavier; Spinelli, Kamryn (The PUMP Journal of Undergraduate Research, 2021)
We provide an explicit construction of a sequence of closed surfaces with uniform bounds on the diameter and on Lp norms of the curvature, but without a positive lower bound on the first non-zero eigenvalue of the Laplacian λ1. This example shows that the assumption of smallness of the Lp norm of the curvature is a necessary condition to derive Lichnerowicz and Zhong-Yang type estimates under integral curvature conditions.