  We provide an explicit construction of a sequence of closed surfaces with
uniform bounds on the diameter and on $L^p$ norms of the curvature, but without
a positive lower bound on the first non-zero eigenvalue of the Laplacian
$\lambda_1$. This example shows that the assumption of smallness of the $L^p$
norm of the curvature is a necessary condition to derive Lichnerowicz and
Zhong-Yang type estimates under integral curvature conditions.
