Bottom of spectra and amenability of coverings

2018/03/20 by Werner Ballmann, Ballmann, Werner, Henrik Matthiesen +3 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · #35P15 #53C99 #58J50 #Dermatological and Skeletal Disorders #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.1803.07353

openalex publication_date 2018/03/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For a Riemannian covering π\colon M₁→ M₀, the bottoms of the spectra of M₀ and M₁ coincide if the covering is amenable. The converse implication does not always hold. Assuming completeness and a lower bound on the Ricci curvature, we obtain a converse under a natural condition on the spectrum of M₀.

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• On the spectral stability of finite coverings

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