Manifolds with bounded integral curvature and no positive eigenvalue lower bounds

2021/03/22 by Connor C. Anderson, Anderson, Connor C., Xavier Ramos Olivé +3
Mathematics · #58J50 (Primary) 53C21 #58J60 (Secondary) #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C21 #msc:58J50 #msc:58J60

paper · pdf · doi:10.48550/arxiv.2103.11970

arxiv created 2021/11/03 · arxiv updated 2021/11/04

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.

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