Manifolds With Bounded Integral Curvature and no Positive Eigenvalue Lower Bounds

Abstract

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 λ₁. 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.