2021/06/03 by Nelia Charalambous, Charalambous, Nelia, Zhiqin Lu +1
Mathematics · #Algebraic Geometry and Number Theory #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Primary: 58J50 #Secondary: 58E30 #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2106.01992
openalex publication_date 2021/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this article we prove that, over complete manifolds of dimension n with vanishing curvature at infinity, the essential spectrum of the Hodge Laplacian on differential k-forms is a connected interval for 0≤ k≤ n. The main idea is to show that large balls of these manifolds, which capture their spectrum, are close in the Gromov-Hausdorff sense to product manifolds. We achieve this by carefully describing the collapsed limits of these balls. Then, via a new generalized version of the classical Weyl criterion, we demonstrate that very rough test forms that we get from the ε-approximation maps can be used to show that the essential spectrum is a connected interval. We also prove that, under a weaker condition where the Ricci curvature is asymptotically nonnegative, the essential spectrum on k-forms is [0,∞), but only for 0≤ k≤ q and n-q ≤ k≤ n for some integer q≥ 1 which depends the structure of the manifolds at infinity.