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arXiv · 2103.11970

Manifolds with bounded integral curvature and no positive eigenvalue lower bounds

Connor C. Anderson·Xavier Ramos Olivé·Kamryn Spinelli
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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 $\lambda_1$. 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.

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Connor C. Anderson, Xavier Ramos Olivé, Kamryn Spinelli. 2021-03-22. Manifolds with bounded integral curvature and no positive eigenvalue lower bounds. https://arxiv.org/abs/2103.11970

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