Manifolds With Bounded Integral Curvature and no Positive Eigenvalue Lower Bounds
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.
Copyright (c) 2021 Connor Anderson, Xavier Ramos Olivé, Kamryn Spinelli
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